A particularly beautiful way of singing the Psalms is when they are set to Anglican chant. We use this form of psalmnody frequently at the parish, and it's always a joy to participate in this style of chanting. Based upon the earlier practice of plainsong, Anglican chant provides a way of singing non-metrical verse using four-part harmony, and it's common to hear it in Anglican cathedrals and parishes with a strong liturgical music program.
Here's a fine example sung by the King's College Choir:
When I was a young theological student studying at Sarum in England, my buddies and I heard the wonderfully funny rendition done by the Master Singers, setting a weather report to Anglican chant:
After hearing that, some of us would have contests to see who could come up with the most bizarre concoction, so one was liable to hear the local police blotter, or portions of C. S. Lewis' "Mere Christianity" sung to Anglican chant. Ah, those carefree student days!
Here's one more from the Master Singers: