The answer to this question depends on which broad system you accept when reading the Bible.
Those systems that see a sharp discontinuity between Old and New Testament (e.g., Dispensationalism, New Covenant Theology, Progressive Covenantalism) will say something like, "The Old Testament laws are done away with in the different era of the New Covenant," or "The OT laws are fulfilled by Christ, who has given us a new, higher ethic." This ethic usually includes nine of the Ten Commandments, excluding the sabbath.
Those systems that see more continuity between Old and New Covenants (usually labeled Covenant Theology as a group) will see continuing applicability of some or all of the OT laws. The approach of John Calvin and the Westminster Standards, two prominent representatives of this group, is to classify laws as moral, ceremonial, or civil, where the moral laws are still in effect but the ceremonial and civil have been fulfilled and abrogated in Christ. (Among the Ten Commandments, the sabbath alone is classified as a ceremonial law and no longer in effect.† The others still hold.) Other mainstream Covenant Theologians hold that all the OT laws are still in effect but apply in different ways (see here and here on the sabbath in particular). While Theonomists and Christian Reconstructionists hold that the whole law is still binding and in effect today.
If you accept one of these over-arching frameworks as the most persuasive, then you have implicitly answered the question, "Do Christians need to keep the Ten Commandments?"
† Calvin says, "[The Fourth Commandment] was a ceremonial precept, Paul clearly teaches, calling it a shadow of these things, the body of which is only Christ. (Colossians 2:17.) But if the outward rest was nothing but a ceremony, the substance of which must be sought in Christ, it now remains to be considered how Christ actually exhibited what was then prefigured...."