Thanks Stu, for introducing Beirut, it sounds a bit folk (like indie folk, if I described it right), I definitely hear the traditional elements too.
I like Le Mystere des voix Bulgares, but it's kind of different from Beirut, like Demonoid said it's centered on the female vocals. It's more of a female choir. I've got a recording of Village Music of Bulgaria, also mainly female vocals, singing with a more nasal voice (I'm guessing that's the more authentic sound?). It's great too, you should check it out.
But they are pretty different from Beirut, hope you like it anyway