I'd say Sigur Ros, but I recommend them to everyone regardless of their tastes. Also whoever pointed you in the direction of Thrice did you an outstanding service, because they're amazing.
dredg, The Dear Hunter and Circa Survive are great bands to get into if you like loud fast music but also like prog. You'd probably also like Brand New and Thursday. Listen to Fugazi and At the Drive-In if you want to get into proper emo. The Mars Volta is a must for any prog fan, and if you get into ATD-i, you'll absolutely want to check out Sparta. Isis is an essential alt-metal/post-metal band. Explosions in the Sky is a great first band to get you instrumental post-rock. Battles will get you into experimental stuff and math rock. I'd also recommend giving Deftones and Queens of the Stone Age a good try if you haven't already.