Much of what we read in the Psalms isn't appropriate for Christians to sing, it's just too violent and hateful against others. David seems to always be in a turmoil because of one enemy or another and he repeatedly calls on God to do some terrible things to them and their families.
Reading the words of Jesus and His Apostles confirms their familiarity with these songs probably from early childhood. They were sung regularly by the Jews of Jesus' day and obviously to some extent by the first disciples of Jesus in the first century and beyond. In the New Testament the Psalms is the most quoted Old Testament book but the emphasis on military victory over one's enemies is absent from the dialogue.
Jesus taught a very different message than what is typically found in the Psalms of David. At the very beginning of His ministry, Jesus preached "You have heard it said, 'You shall love your neighbor and hate your enemy.' But I say to you , Love your enemies" etc. Matthew 5:43 There seems to be no such command anywhere in the Old Testament concerning one's enemies, but Jesus may have been referring to "hate your enemy" as a persistent theme found in the Psalms. At the time of Jesus' ministry, that certainly was the expectation of the Jews. i.e. that God would destroy their enemies and give them military and social superiority again.
After Jesus' resurrection and ascension, the Holy Spirit began His ministry with the Apostles and qualified them to write the New Testament. I think they got the new message and learned to couple it with much that had been sung by their people for a lot of centuries.
David wrote:"Blessed be the Lord, who daily bears us up; God is our salvation" at Psalm 68:19 ESV The context of Jesus' words in the Sermon on the Mount sound a lot like the words of David that had been sung by the Jews for centuries. The Apostles carried the message of Jesus to their generation of Jews and Gentiles and achieved a mighty result.
Another example is Psalm 55:22, "Cast your burden on the Lord, and He will sustain you, He will not permit the righteous to be moved." Peter may have been thinking of this song he had sung many times when he wrote, "Humble yourselves under the mighty hand of God, so that He may exalt you, casting all your anxiety on Him because He cares for you." 1 Peter 5:6-7 Of course singing it in a song or repeating the words in a prayer is pretty simple, but actually taking the action is a much different process. However, singing and praying about it surely helps our spirit do it.
For me, singing the Psalms is something like eating fish with bones in it, you have to learn how to eat around those bones to enjoy the meal. I have learned to pray the Psalms with the full understanding of the times in which they were composed as a way to give glory to Yahweh and encourage the people. There is a huge blessing to be realized from all of the Psalms but we must always read them under the overshadowing of Jesus.
I am blessed today by the words of David and the other Psalmists, they give me a new vision of my Father that thrills my spirit often. Thank you Father for preserving these beautiful words for all generations of your family. HALLELUJAH!