Has anybody played the Linux version of Firewatch on an AMD GPU ? I am just looking to see what kind of performance I can expect. I have an R9 390 myself.

Not exactly what you asked, but someone played the windows version on Linux using wine and gallium-nine:
https://www.youtube.com/watch?v=QgYkPJQsavI

His GPU is much slower than yours, so it should be playable at least. I would be interested in native OpenGL performance as well, though.

Hi, just tested it on HD7850 using latest radeonsi, works good on the first sight. I didn't test too much now, however, the performance is good, but I had some minor glitches in game menu.

Any experience on AMD APUs (integrated graphics)?

I'm sure that I won't shock many people with this news but when I tried to run the game on my old R9 390 using the Catalyst 15.12 drivers it would barely get the loading screen and ran extremely slow. Couldn't even start the game like that. I'm not saying many don't know that Catalyst drivers are poor performance in many games but I thought I would let people know my experience.