Make sure you have at least 11.1, and it was installed right, and you built the kernel module and its being loaded before X starts. Your 2D and 3D acceleration is dead, and OpenCL isn't going to work either. And since you'd be on 11.1 or later, make sure you're using SDK 2.4
Thanks for the tips. I have been a bit busy lately, so it might be a day or two before I get to this. Meanwhile, it's limping along CPU-mining :-/
OK, it's been a bit more than "a day or two" :-)
Recap: I have two Radeon HD 6870s running under Debian.
I ended up wiping the system and re-installing from scratch. I have 11.3 installed, with SDK 2.4. It's not crashing any more, but now I get the error "No OpenCL platforms found".
Any suggestions?