I like Vladimir's suggestion. Here are a few more:
In Modern Greek, you say χρόνια πολλά
http://www.youtube.com/watch?v=jiQEKSuXiB0the ancient form of which might be something like χρόνοι πολλοὶ (γένοιντό) σοι! The ancient word for birthday was ἡ γενέθλιος ἡμέρα, so καλὴ γενέθλιος would work. You could also say καλῶς γενεθλιάζοις. "(may you celebrate your birthday well." How about μακάριος (μακαρία) σὺ γένοισο ἐν τῇ γενεθλίῳ σου? There are always multiple ways of saying something in any language, and this is especially true for Ancient Greek. Maybe just καλὴ ἡ σὴ ἡμέρα!
Anyone else have any ideas?