I have been successful at installing Diablo III on PlayOnLinux in LinuxLite 2.6 64bit,with Nvidia/bumblebee drivers.
Here are the steps: (assuming your video drivers are all setup and working !)
1. Download Diablo 3 client from official Blizzard site
2. To Install Diablo 3 using PlayOnLinux, Open POL and Click Install button.
3. search for diablo, select Diablo III, Click install.
4.Then choose "use a setup file on my computer" (choose/find the setup file you downloaded).
5.Now install Diablo 3 as usual, login to battle.net and download D3 game.
(PlayOnLinux will download wine automatically and configure itself.)
If you are using an Nvidia/hybrid graphics setup-
Go to the desktop shortcut, right click properties, look in the tabs , right click and copy the File Path/Location for the shortcut of Diablo III.
-Then paste it in a terminal (be sure the file name is there too. Then use "optirun" or "primusrun" ie: $ primusrun /FILEPATH_AND_NAME
(I had an issue updating and but i can still play. Apparently its a wine issue from what i found on forums. Not sure if it has been resolved yet.)
I hope that helps, I am trying to get all my gaming done on LinuxLite. (Gaming is the obstacle i have left in order to fully convert over to Linux)