Just reporting back here that I also got it running on Gentoo Linux 64-bit and played all the training games without noticing any bugs and with great performance.

There were some hangups to get it running though:
1) After running battle.net setup wine auto-runs the client and I tried this several times getting a connection error so it would start in "offline mode". However, if I just went into the battle.net directory and started the Launcher directly it worked.
2) When installing Hearthstone the battle.net client would crash. This was fixed thanks to Sinedio's comment above (disabling the dbghelp library in winecfg)
3) The first time I launched the game it was just a black screen and had to force it to close, the second time I launched it I had audio but my display was set very strangely, my monitor was displaying wierd grainy dots like the video mode wasn't set properly and even after closing the game my desktop still appeared this way (fixed by going to tty terminal and back to desktop)
The third time I launched the game it worked like a charm.