A number of bands I like:
Animal Collective - Don't know what genre they play.
Antony & The Johnsons - Dark pop, chamber pop.
Beirut - Indie inspired by Balkan music.
Caspian - Post-rock.
Do Make Say Think - Post-rock.
Efterklang - Post-rock, electronica... I guess? They used to be very experimental. It's usually orchestral though.
Eyes Set to Kill - I listen to pop rock emo stuff once in a while.
For A Minor Reflection - Post-rock. Icelandic.
God Is An Astronaut - Post-rock.
Godspeed You! Black Emperor - Best band ever.
Post-rock.
Gorillaz - You know them.
Halves - Alternative rock.
The Jesus & Mary Chain - Hipster-promoted noise rock, but pretty good.
Kashmir - Danish alternative rock band.
Leonard Cohen - You know him.
Low - Slowcore, very depressing and slowpaced rock.
Mew - Danish indie band and nostalgic to me.
Mogwai - Post-rock, albeit a little metallic at times.
Mono - Post-rock.
Múm - No idea what their genre is called. Check them out. They're good, and Icelandic.
Nyctalgia - Small Swiss post-rock artist - and he's excellent.
Ólafur Arnalds - Another Icelandic artist, he's neoclassical or post-rockish. Take your pick.
paza - 8bit.
Pink Floyd - You know them.
Radiohead - You know them.
Scraps of Tape - Post-rock.
Sigur Rós - Not sure what this band plays, it's post-rock-inspired though. It's one of those bands you're supposed to have heard about if you are interested in modern rock music, many people take it for granted and many people don't. Oh, and they're Icelandic - an excellent island when it comes to music, really, they're overrepresented on my playlist...
The Silver Mt. Zion Memorial Orchestra - A side project with most of the guys from Godspeed You! Black Emperor. They change the band name for like every album they release. Post-rock.
Slint - Hipster-promoted rock, and it's really good.
Slowdive - Shoegaze.
System Of A Down - Guilty pleasure!
This Will Destroy You - Post-rock.
Tortoise - Jazzy post-rock electronica technique and groove nerds. It's awesome.
Trespassers William - Slow, sad shoegaze.
65daysofstatic - Excellent; have only listened to one album, but so many times I can't count. I haven't had the time to check out their other stuffs. Post-rock/electronic something.