Akiko Yano has been a respected figure in innovative music for almost 3 decades, melding jazz and electronics with her quirky songcraft. Somewhat like Bjork, her music may sound off-putting at first but will grow on you.
I came into this thread to recommend Sheena Ringo but it seems you already like her. Back in the day user Aeriyn told me to get that one album of hers with the teacup on it, really glad I did.
In terms of Japanese stuff pretty well known in the US, I second Boris. Great band, start with the album Pink I would say. DJ Krush is great also, hip hop beats with subtle influence of traditional Japanese music.
I just looked up this Shiina Ringo chic, I like her music a lot. You might want to check out a band called Jaurim from Korea, they sort of have a similiar thing going on imo (well with some songs anyway, since they cover a huge range of genres).
Definitely the Yoshida Brothers. Their Japanese albums are more traditional shamisen playing. Their international albums are more along the lines of using a shamisen in rock music.
If you like piano/jazz then Hiromi is for you. You never quite know what to expect. She's the kind of artist that, if you know she's going to be on TV, you MUST record her, as you wont here what she plays ever again. I missed recording her and i'm still kicking myself for it as she was amazing.
This is about as far from mainstream as you're going to get:
Sigh, one of my favorite black metal bands. They incorporate a lot of classical, experimental and avant-garde elements into their music, which makes it really stand out. Mirai Kawashima is a genius...his vocals are quite unique too. Start off with either "Hail Horror Hail" or "Hangman's Hymn - Musikalische Exequien" from their albums.