Coincidentally, I just installed LDD on my, admittedly pretty old, (2011) MacBook Pro and had the same problem. It worked fine on my PC but that it died so I've gone back to the old Mac... yet it used to work fine on there too. Gutted.
It always baffled me why the thumbnails were rendered in 3D anyway - only time I'd notice was when I accidentally rotated one (and then, in a slightly OCD fashion, try to rotate it back exactly how it was before).
Anyway, have you tried an older version?
http://biomediaproject.com/bmp/lmp/games/
I had to delete the LDD folders from ~/Library/Application Support/, ~/Library/Caches/ and ~/Library/Preferences/ as well as the application itself to get it working, but version 4.2.5 does seem to be usable. Yay! I'll probably try the later versions once I've had a play.
Good luck!