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Author's Note: This is an attempt to inform, not to proselytize. (see here)

It is a common misconception that Christians are expected to live within the laws given to Moses in the first few books of the Old Testament. This includes everything from keeping ritual sacrifices, keeping a kosher diet, and sowing a garden in a certain way to wearing blue tassels on our clothes, abstaining from cross-dressing and homosexuality, and so on and so on. This was all well and good for the Israelites, but it was given to them to make their nation prosperous. Because God called them to be his chosen race (for reasons beyond the scope of this node) He provided them with a lengthy guideline on how to grow and stay strong. The Old Testament includes many things: an instruction book on how to build a nation in the ancient world, a foreshadowing of the coming of Christ, some very beautiful poetry, and a remarkable history; but it can never be taken as a binding set of rules for all people and all eternity.

This is hardly a new misconception, in reality it's something the church has been dealing with its entire life. As Paul made his missionary journeys around the ancient world, preaching freedom from the Law of Moses, a group of ex-pharisees (known loosely as the Judaisers) would follow a few days behind him, preaching that Gentiles had to become Jews before they could become Christians. This teaching included circumcision, a kosher lifestyle, and generally following Mosaic Law. When Peter heard of this, he came before the Christian sect of the Pharisees, saying:

"...Brothers, you know that in the early days God chose me to be the one among you through whom the Gentiles would hear the message of the gospel and believe. 8 God, who knows everyone's heart, showed them he approved by giving them the Holy Spirit, just as he did to us. 9 He made no distinction between them and us, because he cleansed their hearts by faith. 10 So why do you test God by putting on the disciples' neck a yoke that neither our ancestors nor we could carry? 11 We certainly believe that it is through the grace of the Lord Jesus Christ that we are saved, just as they are."
--Acts 15:7b-11

The Mosaic law was difficult to follow to the letter. Many times, however, these difficulties arose from how the Pharisees and Saducees interpreted the law. These men were not only the Clergy, but also the Legislative, Executive, and on occasion the Judicial branches of their entire political system. If someone wanted to know if it was kosher to tie a certain type of knot on a certain day, they'd ask one of them (they're also referred to as Lawyers). If someone had a complaint against his neighbor, they'd bring it up with them. This eventually led to a very large, complex system which (by Peter's own admission) was impossible to follow.

Paul says "For by grace you have been saved through faith; and that not of yourselves, it is the gift of God; not of works, so that no one may boast". Salvation comes by grace, not by works. If we could somehow earn our way into heaven by doing enough good deeds, shying away from the bad deeds, there wouldn't be any need for Grace. Paul addresses this issue in his letter to the church in Rome: "For all have sinned and fallen short of the glory of God", continuing "For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord". Now, don't hear me wrong: works are an important part of faith (James even goes so far as to say 'Faith without works is dead'), but it's the Faith that leads to eternal life. Consequently, anyone preaching salvation by way of works (salvation by Mosaic Law, salvation by indulgences, salvation by baptism, etc) is falling into the same trap many pharisees fell into thousands of years ago.

Paul continues: "If you are led by the Spirit, you are not under the Law.", "Therefore no one is to act as your judge in regard to food or drink or in respect to a festival or a new moon or a sabbath day -- things that are a mere shadow of what is to come", and "For sin shall not be master over you, for you are not under Law but under grace". Keep in mind who Paul's talking to: young churches, mostly of a Jewish background. When he speaks of "the Law", he's referring to Mosaic Law. When he speaks of "the law", he referrs to state law. (I say this so nobody thinks I'm claiming Paul says to disobey all laws.). Jesus clearly states in the the Gospel of Matthew that He came not to abolish the law, but to fulfil it. He didn't destroy the Law of Moses, he completed it. He fulfilled the prophesies of the coming messiah, He became the very essence of the suffering servant, (foreshadowed here, here, and here; reinforced here and here.)

So what laws do apply to christians?

"37And He said to him, 'You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.' 38 This is the great and foremost commandment. 39 The second is like it, 'You shall love your neighbor as yourself.' 40 On these two commandments depend the whole Law and the Prophets."
-- Jesus, in Matthew 22:37-40

In short, Everything comes from love.

Further Reading:
Acts 15: 1-12
Romans 12
Romans 13
1 Corinthians 1:17-25
1 Corinthians 13:4-13
2 Corinthians 5:7-9
Ephesians 2: 4
Hebrews 11: 1-2
Pretty much all of Galatians (thanks to xriso for pointing that one out)

m_turner brought up the point of The Seven Laws of Noah (a.k.a. the Noachide Covenant). I feel they, being just as pre-Christian as Mosaic Law, would equally not apply to Christians.