You could throw Frozen River in the category of straight-up drama, but I'd consider it a very low-key thriller. It's not too tense, but enough that kept me on edge, particularly the second half when the plot really started to pay off.
Terrific acting by both female leads. Melissa Leo seems to be getting the most attention, but Misty Upham comes across so natural she almost seems like she was just grabbed from the local area and isn't a professionally trained actress. Also, I'm always a sucker for Michael O'Keefe.
Great script, too. Not cliche at all, which I kept being afraid it was going to veer off into, but it never did.