Praying the Psalms is an incredibly old tradition. The psalms were prayed by Jews long before Christ. Christ himself prayed the psalms. The early Christians immediately began to pray the psalms in light of their significance regarding Christ.
In fact, the praying of psalms is so rich a tradition that in the Liturgy of the Hours, the psalms are prayed several times a day in a continuous cycle.
At every Mass, there is a psalm after the first reading. Many of the antiphons and other texts come from the psalms as well.