I sing the Psalms from the lectionary using the Psalm tones published by the Lutheran Liturgical Prayer Brotherhood. Find the LLPB website here. Find MP3s of (almost) all of the Psalms, in King James Version, here. Each day I usually try to hit all the Psalms from the past seven days and all the Psalms from the upcoming seven days.
The LLPB sings the Psalms in beautiful, a capella Gregorian chant. You can buy their breviary and learn to sing them yourself, or just do like I do and sing along to the MP3s to learn them. Sometimes I have the time to follow along with the text and sing myself, sometimes I just put them on in the background, but even then I find that they are sinking into my heart and memory. The earlier church had a great idea to just sing the Word of God like this. The Psalms were meant to be sung, and singing them with these tunes you can learn to sing along with thousands of others around the world. Sing like the church in the New Testament (not exactly, but closer) and produce and listen to psalms, hymns, and spiritual songs rather than being distracted by mere notes produced by mechanical instruments.
Moreover, as you sing and pray the Psalms, it is like having the Holy Spirit guide you, as if an expert teacher guides your fingers while you learn a complicated task, until your habits are well-formed and grow like His. We know not how to pray as we ought, but the Spirit intercedes for us.