I wish I had more specific help for you, but I run Alice on Ubuntu and have been doing so for a half year or so. On Ubuntu you can start Alice from the command line, but clicking on the run-alice file and then choosing "run in terminal" is faster.
I have found that setting desktop effects to NONE in Ubuntu helps when trying to run some Java programs. Also, make sure you have all the necessary Java files installed. You may want to try Sun's JDK instead of using Open JDK.
I really hope Linux compatibility is a focus for Alice 3.0...