The Armstrong Church of God used to sing the Psalms directly. Herbert Armstrong's brother Dwight Armstrong wrote most of the music in their hymnal. This approach makes sense. God gave us the Psalms to sing. What good is it singing all sorts of man-made hymns, when we ignore the hymns God himself gave us? Before Herbie and Dwight came along, the Scottish Covenanters had the same idea. They published the Scottish Psalter. David Fister took the time to put the sheet music scores and the texts up on the web. You can read it here: Scottish Psalter for singing. David Fister also has a great website, Commandments of יהוה.
The Psalms in the Psalter are slightly different from the text of the Psalms as given in the King James version of the Bible. The words were rearranged and altered a little bit so that they would fit a regular meter. This allowed the Covenanters to use simple, regular tunes to sing the Psalms. Until we all learn Hebrew and sing the Psalms to their original melodies, this is as good as it gets. The changes to the words are not major, nor do they change the meaning appreciably. The Bay Area Psalm book is also good.
I hope to incorporate the Scottish Psalter into the Reactor Core Hymn Repository soon.