I searched and didn't see this posted and while this may not be an exploit, I figured I put this in the guide section. Diablo 3 is the first new Blizzard game that doesn't allow game in the background for some weird reason. I personally miss having the game in the background, because I stay logged into battle.net all day. This way my friends can message me if they need something, farming partner, help with a kill, chest to loot etc etc etc. With sound in the background, I can hear the ding tone anytime someone sends me a message. This lets me browse the web, watch a movie, and not miss any messages.
On to the tip, tested and confirmed working on Windows 7. Have not tried this on other OSes.
#1 Under display, Screen MUST be set to either "Windowed" or "Fullscreen Windowed". Must be those two or it will not work.
#2 Instead of alt-tabbing or minimizing the window to do other things on the computer, hit the windows key while in game. This should bring up the task bar at the bottom. Then select anything from the task bar and you will jump to that but keep the sound from Diablo 3 in the background.
#3 Personal tip, I like to pick the show desktop button in the quick launch bar, but that's a personal preference.