I've only seen the first season of the HBO series, and not even all of it, so keep that it mind. That said, the only characters I felt any empathy for were the Starks (including Jon Snow).
If you've got any idea what GRRM's series is about, you know that the Starks get metaphorically defecated upon. Frequently. Because I'm not averse to reading spoilers, I read ahead a bit on plot synopses. And thank goodness I did, because after what happens to Ned Stark, I can honestly say that GRRM can write whatever he wants, but I will neither pay for nor spend my time on anything that he's touched. I believe I actually said out loud (yes I was talking to my computer screen) "F- you, George. F- you straight to hell." And I haven't looked back since.
I simply don't like any other characters enough to care what happens to them. And I don't care what kind of story GRRM is trying to tell, because he kills/cripples/maims/otherwise completely screws over House Stark as his first order of business. It's entirely possible that he stops screwing with them at some point, but I'd bet that it's only because he's killed them all.
Tyrion is an interesting character and all, but he's hardly enough to keep me invested in the story.