Rin' feat. alan - Sennen no Niji
Actually, anything by Rin' would fit, though alot of their stuff is just instrumental if that's ok. I can upload some of their albums if you like.
Also the acoustic orchestra version of "I am..." (from Ayu-mi-x 4 I think?) is very Japanese sounding too (and awesome).
Some Shiina Ringo too (Tsumiki Asobi and Shuukyou come to mind. I'm pretty sure there's more, but these are off the top of my head).