I've tried using print screen, but it seems to grab the window behind Diablo rather than the game. Is there some in-game hotkey to take a screenshot, or some other method that will work?
Hit Print Screen. The screenshots are automatically saved as JPGs in your
[User]\Documents\Diablo III\Screenshots folder. You can rebind this shortcut in settings as LessPop and Augustus note.
You do not need to paste an image into a graphics program (e.g. Paint) and save that, if you do, you will get nothing/garbage as you note.
There's always Fraps. The free version only takes BMP though.