Not all Christians did. The very earliest Christians were the Jewish followers of Jesus living in Roman Palestine, but within a few decades after the death of Jesus, the number of Jewish Christians was probably surpassed by gentile converts in cities throughout the Roman empire (in Greece, Asia Minor, etc.), due to recruitment by the apostle Paul and other traveling preachers. While Jewish believers in Jesus probably still celebrated Jewish holidays, gentile Christians did not.

In fact, one of the major early controversies in the church appears to be exactly the question of whether converts to Christianity must also essentially become Jewish, which would include things like male circumcision, following dietary laws, and celebrating Jewish festivals. We are told that Jesus' disciple Peter, who led the early Jerusalem church, argued that Christians must also become Jews, while the apostle Paul believed that this was not necessary, and that all that was required was faith and baptism in the name of Jesus.

Both schools of thought had their adherents (a more Jewish-leaning version of Christianity is evident in the theology of the Gospel of Matthew), but over time the version of Christianity preached by Paul, which de-emphasized pre-Christian Jewish religious traditions, gained far more followers and became the norm for the emergent Catholic Church.

Edit: For a good and very accessible work of popular history on this subject, check out Bart Ehrman's book "The Triumph of Christianity." I know, I know, it's an awful title that makes it sound like a gloating work of Christian apologetics, but it's actually just a pretty straightforward secular account of the history and growth of the early Church.