If you are into Christian Koine, there are a number of songs written out and a few a Capella at http://www.letsreadgreek.com/songs
. There's even an Ode to Brett Farve sung to the Clash's Should I go Or Should I stay. I've got a number of other secular ones like Your So Vain, Like a Rolling Stone, and some others, but they are not at that site. It's hard getting the accents to match the rhythm -- many reworks, word changes, form changes, line flips, simplifications (English has many 1-2 syllable words) and alterations are sometimes necessary.
And then there is the pronunciation scheme -- I use Restored Koine (1st Century AD) pronunciation -- I wouldn't want to use Erasmian -- Restored Attic or Modern would work also. But with modern you have some vocalic syncracisms (ὑμεῖς = ἡμεῖς), κτλ.
The Classical Review, Fall 2009, is also supposed to have some songs in Greek. This is one of my favorite things to do, so I should keep a running list of all the songs I find and where they are. So I'll keep my eyes open. Until I can get a good singer and musician/pianist together, the best thing is to record yourself singing them and listen to them. My a Capella stuff is not for the weak of heart (I can hold a tune and thats about all).