I finally managed to run Team Fortress 2 on my system, without installing proprietary drivers.
64-bit Linux Mint 13 MATE (corresponding to Ubuntu 12.04)
AMD Radeon R600 video card
(System information by Steam
To summarize what I had to do:
* install a package for S3TC texture compression
* compile and install new versions of libdrm2 and Mesa
* run Steam in an environment that sets localization to "C" and uses the newly compiled libraries
I don't know yet how to enable display of the frame rate, but I'd estimate it runs about 10-15FPS on my system.
So I'd say it's not quite ready yet to be run out-of-the-box on Ubuntu 12.04 or Mint 13, but with some "Linux hacking" it can be made to work. Maybe it will install smoother on newer distributions, if they already have a sufficiently new Mesa version.
To kick-start people who have a similar system, I prepared this archive
which contains my compiled binaries for libdrm2 and Mesa.
If you want more information, please reply.