Not sure how definitive an answer I can provide to this question, but thought I'd contribute a few thoughts that came to mind.
The "law," or "law of Moses," was codified by the Rabbi's in ancient times into the mitzvot, a collection of 613 laws compiled from a close reading of the Pentateuch.
The mitzvot consists of "positive" (ie "do this") and "negative" ("do not do this") requirements, and is traditionally divided into three basic categories:
- The "ceremonial law"
- The "dietary law"
- The "moral law"
Christians, believing that Christ is the "fulfillment of the law," believe that his atonement rendered the ceremonial and dietary laws void. Peter's vision in Acts 10:9-15 is sometimes offered as a proof that the Old Testament dietary law does not apply to Christians.
(Note that the requirements in the "moral law" category still apply, because, AFAIK, they are consistent with -- and, essentially, the basis for -- Christ's teachings about how to love others).
Edit: A link to the mitzvot follows -- however, please note that I am no scholar of Judaism, and I don't know if the site is credible or not. It's apparently a personal project maintained by a Tracey R. Rich, and is currently ranked fourth for a Google search of "mitzvot." "Judaism 101" link. Their list uses a different set of classifications, which the page says was created by the site's editor. Some examples of the Jewish dietary laws can be found in items 143-169.
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1 Corinthians 10 is also germane to the question; see v. 14-31, esp. v. 25-30. In 1Cor 10, Paul emphasizes that what's important to God is Christians' love for one another, rather than their legalistic compliance with things like dietary requirements. See v. 28-30. This is consistent with the Christian concept of God's grace, and its priority over legalism. The section headings in the NIV translation are helpful in this chapter.
Also, in Acts 15, Luke quotes James as elaborating the requirements that would apply to Gentiles who wanted to join the church. In verse 20, he says that they are not to eat meat from animals that had been strangled, and they are not to drink blood. I think I recall hearing at some point that strangulation of animals was a common practice in pagan ceremonies of the time, and that the overwhelming majority of the meat that could be purchased in the market in cities like Corinth had been slaughtered in such ceremonies.
I wish I had some references to good commentaries to substantiate more of this response for you, but I'm a bit out of my depth here. And I'm not familiar with the teachings & traditions of the Eastern Orthodox church, so I can't address that element. FWIW, I hope this response provides something of a starting point for further inquiry. Cheers.
Edit: Also useful here is Paul's discussion of vegetarianism in Romans 14.
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