The Prince of Nothing Trilogy by Scott Bakker (starting with The Darkness That Comes Before). Political intrigue and a lot of dark, grim things happen to the characters. He lacks GRRM's sense of pacing or humour, however (it probably should have been a duology).
Memory, Sorrow and Thorn by Tad Williams (The Dragonbone Chair). The series that helped inspire GRRM to write his series.
The Monarchies of God by Paul Kearney (Hawkwood's Voyage). A similar mix of politics, battles and great characters. Out of print for some years, but thankfully being republished next summer in one omnibus volume.
The Empire Trilogy by Raymond E. Feist and Janny Wurts (Daughter of the Empire). An interestingly different world (based on Oriental culture) with a ton of politics and some excellent characters. Overlong though. It may help to read Feist's Magician first.
The First Law Trilogy by Joe Abercrombie (The Blade Itself). A somewhat traditional first volume is nicely subverted in the sequel. Some of the characters, such as Logen and especially Glokta, are worthy of comparison with ASoIaF.
The Gentleman ******* by Scott Lynch (The Lies of Locke Lamora). Lynch's writing style is very close to GRRM, although he prefers lower-level stories about criminals and con-merchants to political machinations.
The Kingkiller Chronicle by Patrick Rothfuss (The Name of the Wind). More simplistic and cliched than ASoIaF, but the writing is good and the characters are interesting.
The Kingdoms of Thorn and Bone by J. Gregory Keyes (The Briar King). Described as GRRM-lite by some.
The Sword of Shadows by J.V. Jones (A Cavern of Black Ice). Exceptionally good, dark, gritty fantasy set in snow-covered northern lands with political intrigue and superb characters. A close match for GRRM. You may wish to read The Book of Words Trilogy (The Baker's Boy) first, which is more simplistic and aimed at younger readers, but is set in the same world and shares some characters.
The Helliconia Trilogy by Brian W. Aldiss (Helliconia Spring). Actually not very much like ASoIaF at all, but an SF treatment of the idea of living on a planet where the seasons last a long time (centuries, in this case).
The Gap Series by Stephen R. Donaldson (The Gap into Conflict: The Real Story). The closest SF series to ASoIaF. Gritty, with incredibly well-drawn characters.
Crown of Stars by Kate Elliott (King's Dragon). Like ASoIaF, this is a (complete, thankfully) seven-volume series set in a world closely based on Medieval Europe. Not as well written and it's too long, but the series has a decent ending that just about worth it, despite lengthy plot deviations.