Which Ten Commandments do the Christians follow the one in Exodus 20 or Exodus 34 or Deuteronomy 5?
Deuteronomy 5 repeats them
Ex 34 just lays out some of the Mosaic Law.
Christians are not under the Ten Commandments, as such, though the principals are still very valid. The Ten Commandments were part of the Law given to Moses to judge the nation of Israel, but it's important to realize that one could NEVER just 'obey the ten commandments' and be approved by God. Paul said that if a person obeyed part of the law, they were "under obligation to perform the WHOLE law', sacrifices and all. (Galatians 5:3-6)
Knowing the REASON for the Law will help to answer your question:
Galatians 3:19 tells us that the Law was added 'to make transgressions manifest' and verses 23-24 says that the Law was a 'tutor leading to Christ'. Though the principles involved reveal God's feelings on things and help us all to know who he is, the LAW, including the Ten Commandments, was written expressly to and FOR the descendants of Jacob...the Israelites.(Psalms 147:19-20)
The Mosaic Law taught the Israelites that they could not be perfect. They could not show PERFECT love for God or for their neighbor, or carry out God's requirements perfectly. It taught them what sin was and that, because they were imperfect, they needed help to be acceptable to God. They needed a Messiah. This is why sacrifices were necessary. Animal sacrifices covered their sins on a regular basis and pointed toward the coming messiah who would provide a FINAL sacrifice, once for all time.(Hebrews 7:26&27)
The Law protected the Isrealites and taught them God's righteous principles and would prepare them for the coming of the Messiah and guide them to him.
When the messiah came and fulfilled the Law, (by keeping it perfectly and giving his life as the final sacrifice) ,'The Law' (including the Ten Commandments) would END. (Colosians 2:13 & 14) (Galatians 3:23-25)
Jesus arrived at the prescribed time and filled the role as messiah, or Christ.
As a Jew, he was born as a human being 'under law' (Galatians 4:4). When he lived his life perfectly, and perfectly kept NOT just the 10 Commandments, but the whole Mosaic Law in all it's detail, unlike any other human being had ever done, he fulfilled the law (Luke 24:44).
When he died completely faithful, never having broken God's law in any way, he ended the need for the written Law code (Romans 10:4) and a NEW LAW would be in force, the 'New Covenant' spoken of at Hebrews 8:13, which made the OLD one, the Mosaic Law, obsolete. (Matthew 5:17) (Ephesians 2:15)
The law to Christians was to be 'written on their hearts' (Jeremiah 31:33) and instead of being burdened with many laws to follow physically, we would closely follow God's principles in our hearts, accepting Jesus' sacrifice to make up for where we unintentionally fell short.
Infact the Bible indicates that if we still try to please God by practicing any part of the Mosaic law, we are denying what Christ did for us. Christians do NOT live by 'Law' (that is, the law code given to Moses), we live by FAITH, trusting that Jesus' sacrifice is what saves us, not following a written law code. "Galatians 5:4
The principals set out in the ten commandments, however, are still valid, and incorporated into only TWO LAWS:
Matthew 22:37- 40
"Love God with your whole heart, soul and mind,
and your neighbor as yourself.
Upon these, the whole Law hangs."
(Romans 6:14) (Colossians 2:13&14) (Romans 7:6) (Galatians 5:18)
Christians are under the law of LOVE (John 13:34&35) which encompasses and goes far beyond the Ten Commandments, involving NOT just actions, but heart and MOTIVE. (Matt. 5:21, 22, 27-30)
On the other hand, Christians should not follow the Law simply to follow the Law, thinking that doing so will win favor from God.
The Mosaic Law, though, went far beyond these ten line items. The book of Deuteronomy summarizes the full Mosaic Law as given by God to the people of Israel through Moses. As time went on, the Jewish priests would expand upon the law, presumably to clarify various questions over situations, but more often than not limiting people and expanding the power of the high priests.
When God's Law was given, He was setting a standard that He knew nobody could follow. Paul admits that all have sinned and fallen short of the glory of God (Romans 3:23). Theoretically one could follow the entire law with his whole heart � not just in deed � and get to Heaven. The problem is, one strike and you're out. If you have ever lied, even once, you're barred from Heaven by the Law. The Law did provide a way to "cover" one's sins, but it wouldn't change the heart. That's why God sent Jesus, His only begotten Son, not to condemn anyone, but to save the world (John 3:16-18). We cannot do it on our own; we need Him.
Christians are called to do what God wants them to do, but not simply as mechanical beings. We do what God wants because we trust that He knows better than we do about what's best for us. We do it because we love Him, because He first loved us. For the Christian, the Ten Commandments, or even the entire Mosaic Law, isn't a checklist to present to Heaven's gatekeeper as an admission ticket or a test we must pass. The Law is part of God's revelation of what is best for us, and we do it because we want to.
I would just like to add one other comment about the Ten Commandments. Christians today do not keep the Sabbath Day as it is commanded to the Jews. Christians meet on Sunday, which is the first day of the week, as commanded for the New Testament christian.