Why do most Christians not follow the Talmud? If I'm not misunderstanding anything, Jews usually follow the Tanakh (more or less the Old Testament) and the Talmud, but definitely not the New Testament, whereas Christians follow the Old and New Testaments but not the Talmud. So, among those who believe in the Old Testament, people usually follow the Talmud if and only if they do not believe in the New Testament.

Why is this? Are the Talmud and the New Testament incompatible, or is the combination simply unfashionable or something?
The Talmud "takes the form of a record of rabbinic discussions pertaining to Jewish law, ethics, philosophy, customs and history." (Wikipedia)  It was written by ancient Rabbis as discussions and interpretations of the scriptures of the Tanakh (Old Testament).

The New Testament teaches that Scripture must be given by inspiration of God.  As the Talmud was not, and does not claim to be, inspired writing, it is not considered authoritative by Christians.

Also, many of the passages of the Talmud are dedicated to interpretations of various provisions of the Law of Moses, which Christians believe were fulfilled in Christ and no longer apply, which makes a great deal of the Talmud not only non-authoritative but also irrelevant, from a Christian perspective.