Not sure exactly how obscure a lot of these are, but here are some of my favorites. : D
I Monster - Literally any of their albums. They're my favorite band hands down, mostly creating more weird, pop-y music. (I think the term was "trip-hop") The best starter album would probably be "Never odd or even" as it's the easiest to find.
Venetian Snares - For when you're looking for something very noise-y and weird. I'm personally fond of the album "printf<"shiver in eternal darkness / n">;" though I really love most all of their work.
Godspeed You! Black Emperor - You've probably heard of this one before if you're into "obscure" music, but they're still great, and I feel the need to recommend them. It's music you kind of have to be in the mood for, but man, when that mood hits it's perfect.
"From the Ground" by Heather Woods Broderick - I've only listened to the one album but it's great. Softer, and with a mildly unsettling undertone, it's really pretty music.
A Silver Mt. Zion - Great, beautiful music. No idea how obscure they are, if they're obscure at all, but their music is gorgeous.