What meats are we, as members of the Church of God (7th Day), commanded to refrain from?
Key Verse:  Ye shall therefore put difference between clean beasts and unclean, and between unclean fowls and clean: and ye shall not make your souls abominable by beast, or by fowl, or by any manner of living thing that creepeth on the ground, which I have separated from you as unclean.  [Leviticus 20:25]
Did God make a distinction between different types of meat?
In the story of Noah and the Ark, God gave Noah specific instructions about the animals he was to bring into the Ark.
Of every clean beast thou shalt take to thee by sevens, the male and his female: and of beasts that are not clean by two, the male and his female.  [Genesis 7:2]
Of clean beasts, and of beasts that are not clean, and of fowls, and of every thing that creepeth upon the earth,  There went in two and two unto Noah into the ark, the male and the female, as God had commanded Noah.  [Genesis 7:9]
God made a distinction between the animals and divided them into two categories:  clean and not clean.  Noah understood this distinction and followed God's instructions.  A further explaination of which animals fell into the clean category and which fell into the unclean category can be found in Leviticus.
And the LORD spake unto Moses and to Aaron, saying unto them,  Speak unto the children of Israel, saying, These are the beasts which ye shall eat among all the beasts that are on the earth.  Whatsoever parteth the hoof, and is clovenfooted, and cheweth the cud, among the beasts, that shall ye eat.  Nevertheless these shall ye not eat of them that chew the cud, or of them that divide the hoof: as the camel, because he cheweth the cud, but divideth not the hoof; he is unclean unto you.  And the coney, because he cheweth the cud, but divideth not the hoof; he is unclean unto you.  And the hare, because he cheweth the cud, but divideth not the hoof; he is unclean unto you.  And the swine, though he divide the hoof, and be clovenfooted, yet he cheweth not the cud; he is unclean to you.  Of their flesh shall ye not eat, and their carcase shall ye not touch; they are unclean to you.  These shall ye eat of all that are in the waters: whatsoever hath fins and scales in the waters, in the seas, and in the rivers, them shall ye eat.  And all that have not fins and scales in the seas, and in the rivers, of all that move in the waters, and of any living thing which is in the waters, they shall be an abomination unto you:  They shall be even an abomination unto you; ye shall not eat of their flesh, but ye shall have their carcases in abomination.  Whatsoever hath no fins nor scales in the waters, that shall be an abomination unto you.  And these are they which ye shall have in abomination among the fowls; they shall not be eaten, they are an abomination: the eagle, and the ossifrage, and the osprey,  And the vulture, and the kite after his kind;  Every raven after his kind;  And the owl, and the night hawk, and the cuckoo, and the hawk after his kind,  And the little owl, and the cormorant, and the great owl,  And the swan, and the pelican, and the gier eagle,  And the stork, the heron after her kind, and the lapwing, and the bat.  All fowls that creep, going upon all four, shall be an abomination unto you.  Yet these may ye eat of every flying creeping thing that goeth upon all four, which have legs above their feet, to leap withal upon the earth;  Even these of them ye may eat; the locust after his kind, and the bald locust after his kind, and the beetle after his kind, and the grasshopper after his kind.  But all other flying creeping things, which have four feet, shall be an abomination unto you.  And for these ye shall be unclean: whosoever toucheth the carcase of them shall be unclean until the even.  And whosoever beareth ought of the carcase of them shall wash his clothes, and be unclean until the even.  The carcases of every beast which divideth the hoof, and is not clovenfooted, nor cheweth the cud, are unclean unto you: every one that toucheth them shall be unclean.  And whatsoever goeth upon his paws, among all manner of beasts that go on all four, those are unclean unto you: whoso toucheth their carcase shall be unclean until the even.  And he that beareth the carcase of them shall wash his clothes, and be unclean until the even: they are unclean unto you.  These also shall be unclean unto you among the creeping things that creep upon the earth; the weasel, and the mouse, and the tortoise after his kind,  And the ferret, and the chameleon, and the lizard, and the snail, and the mole.  These are unclean to you among all that creep: whosoever doth touch them, when they be dead, shall be unclean until the even.  [Leviticus 11:1-31]
This is the law of the beasts, and of the fowl, and of every living creature that moveth in the waters, and of every creature that creepeth upon the earth:  To make a difference between the unclean and the clean, and between the beast that may be eaten and the beast that may not be eaten.  [Leviticus 11:46-47]
The clean animals were given to man for food, while the unclean animals are not to be eaten.

The profit Isaiah reinforces the need to abstain from eating unclean meats in the prophesy given to him concerning the return of the Lord.

For, behold, the LORD will come with fire, and with his chariots like a whirlwind, to render his anger with fury, and his rebuke with flames of fire.  For by fire and by his sword will the LORD plead with all flesh: and the slain of the LORD shall be many.  They that sanctify themselves, and purify themselves in the gardens behind one tree in the midst, eating swine's flesh, and the abomination, and the mouse, shall be consumed together, saith the LORD.  [Isaiah 66:15-17]

These verses state very plainly that the Lord will destroy those that purify themselves by eating swine's flesh.  Clearly, the Lord wants us to abstain from eating these meats.
Did Jesus do away with the law of the clean and the unclean?
To answer this question we need to study the Priesthood under which we live.  When Jesus died for our sins on the cross, he did away with the "law contained in ordinances" or the "book of the law" and was established as a high priest after the order of Melchisedec.  The law that Jesus did away with was the added law that was given to the children of Israel under the Levitical priesthood because of transgression.  However, the laws that were established under the Melchisedec priesthood before the Levitical laws were given still remain in effect today.  It is true that the explaination of the clean and unclean animals was only fully laid out within the Levitical law.  However, the reference to the clean and not clean animals in the story of Noah clearly indicates that this law was understood and in effect under the Melchisedec priesthood in the time of Noah.  Therefore, this law remains in effect for us today.

Much of the confusion concerning the law of the clean and the unclean arises from letters Paul wrote to the Roman and Corinthian brethren.  Paul writes:
Howbeit there is not in every man that knowledge: for some with conscience of the idol unto this hour eat it as a thing offered unto an idol; and their conscience being weak is defiled.  But meat commendeth us not to God: for neither, if we eat, are we the better; neither, if we eat not, are we the worse.  But take heed lest by any means this liberty of yours become a stumblingblock to them that are weak.  For if any man see thee which hast knowledge sit at meat in the idol's temple, shall not the conscience of him which is weak be emboldened to eat those things which are offered to idols;  And through thy knowledge shall the weak brother perish, for whom Christ died?  But when ye sin so against the brethren, and wound their weak conscience, ye sin against Christ.  Wherefore, if meat make my brother to offend, I will eat no flesh while the world standeth, lest I make my brother to offend.  [1 Corinthians 8:7-13]
Let us not therefore judge one another any more: but judge this rather, that no man put a stumblingblock or an occasion to fall in his brother's way.  I know, and am persuaded by the Lord Jesus, that there is nothing unclean of itself: but to him that esteemeth any thing to be unclean, to him it is unclean.  But if thy brother be grieved with thy meat, now walkest thou not charitably. Destroy not him with thy meat, for whom Christ died.  Let not then your good be evil spoken of:  For the kingdom of God is not meat and drink; but righteousness, and peace, and joy in the Holy Ghost.  For he that in these things serveth Christ is acceptable to God, and approved of men.  Let us therefore follow after the things which make for peace, and things wherewith one may edify another.  For meat destroy not the work of God. All things indeed are pure; but it is evil for that man who eateth with offence.  [Romans 14:13-20]
These verses seem to indicate that nothing is unclean unless man makes it unclean.  However, these verses must be taken in context.  During this time there was a great debate within the church as to whether the meat from offerings to idols could be eaten (this was prohibited under the Levitical law).  This debate is referenced in the 15th chapter of Acts.
For it seemed good to the Holy Ghost, and to us, to lay upon you no greater burden than these necessary things;  That ye abstain from meats offered to idols, and from blood, and from things strangled, and from fornication: from which if ye keep yourselves, ye shall do well. Fare ye well.  [Acts 15:28-29]
The law of the clean and the unclean was never being brought into question here.  These verses are references to this debate concerning the meats offered to idols.

Another source of confusion towards this issue can be found in the gospel of Mark.  In speaking with the Pharisees and scribes, Jesus speaks the following words.
And he saith unto them, Are ye so without understanding also? Do ye not perceive, that whatsoever thing from without entereth into the man, it cannot defile him;  Because it entereth not into his heart, but into the belly, and goeth out into the draught, purging all meats?  And he said, That which cometh out of the man, that defileth the man.  [Mark 7:18-20]
These verses seem to indicate that a man can eat anything he wants and it will not defile him.  Again these verses must be taken in context regarding the question at hand.
Then the Pharisees and scribes asked him, Why walk not thy disciples according to the tradition of the elders, but eat bread with unwashen hands?  [Mark 7:5]
Howbeit in vain do they worship me, teaching for doctrines the commandments of men. For laying aside the commandment of God, ye hold the tradition of men, as the washing of pots and cups: and many other such like things ye do. And he said unto them, Full well ye reject the commandment of God, that ye may keep your own tradition.  [Mark 7:7-9]

The discussion Jesus had with the Pharisees and scribes at this time was concerning the washing of hands (and other traditional washings) and not the law of the clean and the unclean.  Within these discussions Jesus admonishes the Pharisees for putting aside the commandments of God.  Why then would Jesus put aside a commandment of God at this point in time when he had just finished criticizing the Pharisees for doing the very same thing?  Jesus's remarks on this occasion only demonstrate that the Jewish traditions dealing with diverse washings need not be followed.

A third source of confusion on this subject can be found in Paul's letter to the Colossians.
Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross;  And having spoiled principalities and powers, he made a show of them openly, triumphing over them in it.  Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days:  [Colossians 2:14-16]
The key to understanding this set of verses is the first clause, "Blotting out the handwritting of ordinances that were against us".  These verses are a reference (with respect to meats) to the Levitical laws dealing with the handling and washing of meats, not the Melchisedec law that sets the commandments for what meats can and cannot be eaten.

A final source of confusion deals with the vision Peter had while staying with Simon the tanner in Joppa.
On the morrow, as they went on their journey, and drew nigh unto the city, Peter went up upon the housetop to pray about the sixth hour:  And he became very hungry, and would have eaten: but while they made ready, he fell into a trance,  And saw heaven opened, and a certain vessel descending unto him, as it had been a great sheet knit at the four corners, and let down to the earth:  Wherein were all manner of fourfooted beasts of the earth, and wild beasts, and creeping things, and fowls of the air.  And there came a voice to him, Rise, Peter; kill, and eat.  But Peter said, Not so, Lord; for I have never eaten any thing that is common or unclean.  And the voice spake unto him again the second time, What God hath cleansed, that call not thou common.  This was done thrice: and the vessel was received up again into heaven.  [Acts 10:9-16]
We know the interpretation of this vision as Peter himself stated it in verse 28:
And he said unto them, Ye know how that it is an unlawful thing for a man that is a Jew to keep company, or come unto one of another nation; but God hath showed me that I should not call any man common or unclean.  [Acts 10:28]
The dream was a message to Peter that he should preach unto the Gentiles as well as the Jews, not a message that he should eat unclean meats.  In fact Peter's very words from this passage attest to the fact he had never eaten any unclean meat.  Peter kept the law of the clean and the unclean throughout his life.

Jesus did not cleanse all meats and do away with the law of the clean and the unclean.  If we take in context each of these previous verses, their meanings become clear and fit well with our understanding that Jesus did away with the Levitical law and only the Levitical law.  The Melchisedec law of clean and unclean meats is still in effect today.

Click on the following link for a more detailed discussion of The Law of the Clean and the Unclean.
From the time of Noah, God has commanded that members of his church refrain from eating unclean meats.  Only the meats designated as clean are to be eaten.  God's commandment concerning these meats was established under the Melchisedec priesthood and remains in effect today.  

The web site of The Israel of God Research Counsel contains a good listing of common commercial products which do and do not contain unclean meats.