The Ten Commandments, which are found in the Old Testament book of Deuteronomy, and elsewhere throughout the Bible, are part of the covenant God made with the Israelites at Horeb; they are a part of Judaism.1 We, also, know this covenant as the Old Covenant.
In considering whether, or not, Christians are obligated to follow all of the Ten Commandments, it's important to understand that Christianity is not a continuation of Judaism; although the two are closely related.
The Christian Church (the body of Christ) is built on the foundation of the apostles and prophets, with Christ Jesus Himself as the chief cornerstone.2 In fact, when Jesus asked His disciples, "Who do people say the Son of Man is?" the apostle Peter replied, "You are the Christ, the Son of the living God." To this Jesus said, "Blessed are you, Simon son of Jonah, for this was not revealed to you by man, but by my Father in heaven. And I tell you that you are Peter, and on this rock I will build My church, and the gates of Hades will not overcome it..." 3
This being so, then how does the Moral Law (the Ten Commandments) fit into the scheme of things under the New Covenant, the covenant which the Christian Church participates in; the covenant which replaces the Old Covenant ? Are we, as Christians, free to disobey the Ten Commandments or are we obligated to follow all of them?
Although Christians are not under the Law of Moses, we are not free to commit adultery or murder etc. as prohibited by the Ten Commandments. All of the Ten Commandments are biblical principles found in the New Testament of the Bible with the exception of the seventh day Sabbath. It has been the custom of Christians to observe the Lord's Day on the first day of the week in remembrance of the Lord Christ Jesus' resurrection from the grave on the first day of the week. Examples of early Christian churches doing this (which are mentioned in the Bible) are the church in Troas, the church in Corinth, and the churches in Galatia.4
1 (Deuteronomy 5)
2 (Ephesians 2:20)
3 (Matthew 16:17-18)
4 (Acts 20:7; 1 Corinthians 16:1-2)