Diablo3 shipped in a state of almost working in wine. Wine however needs some patches to install and run the game. For Ubuntu a PPA exists with packages but it only has builds for precise which are not installable on oneiric-level distros such as Mint and Kubuntu. So I set about to build my own patched wine -and while I was at it, to build a proper debian package. I used the Ubuntu Wine team's PPA as a baseline, and added the needed patches and built the package.
With this wine package I am successfully playing Diablo3 on Linux Mint. Pretty impressive for a game that only launched less than 24 hours ago. To run the game first install the package (link at the bottom of the post). Please note that I didn't write any of the patches here, I just integrated them with a debian package.
Then you need to run: winetricks vcrun2008
Now you can install the game. If you are using the online installer, it should just work (I haven't fully tested this). If you are using the DVD installer there is an extra step. Insert your DVD and open it to browse, then open a terminal and run:
sudo mount -o remount,unhide /media/D3C1.0.0
wine "Diablo III Setup.exe"
The installer should now run and you can install the game. The included wine also has a patch to fix problems with screen switching and selection. The game seems to work perfectly for me.
The package conflicts with wine1.3 as provided by LinuxMint native. If you get errors on this regard remove the wine and wine1.3 packages first.
sudo apt-get remove wine wine1.3
This may remove some packages that depend on wine (such as playonlinux), so make a note of them, you'll be able to reinstall them after you install the download.