Band-Ikue Asazaki
Song: Obokuri-Eeumi
Genre- Japanese (Is that a genre?). Soundtrack from "Samurai Champloo".

The girl's voice takes a while to get used to, but the song is so beautifully sad.

https://www.youtube.com/watch?v=bEQtkLNTmRs&list=FLgWGp9DeQJWgK1c1fJsK8kg&index=2