I have really thick, curly hair, and since it's been hard at times to even find a good person back home who wasn't overwhelmed with my hair, I was very nervous about getting my hair cut in Korea....I put it off for a very long time and got by on some self-trimming and waited until I went home to visit my stylist back there for a real cut.
But this year, I had to do something about it....I can't personally recommend any place in Seoul (I'm sure there are lots of places, and if you search "curly hair" on this site you will find other posts about it), but if you're willing to travel at all, I can recommend someone outside of Seoul. I went to a place called Miss Piggy's in Songtan (just south of Osan on Line 1). She's Korean but speaks English, and she is right by the Osan Air Base so she works with foreigners all the time. I just needed a basic trim, but she did a great job, and best of all, it was insanely cheap...only 10,000 won :) Her number is 010-7762-2358.