I think I'm having the same problem you are. I haven't checked the logs or anything but my launcher just loads with FML and then that window disappears and just shows a white screen without showing the Mojang logo. I also use Ubuntu 12.04, but I use the 64 bit version, according to the system details. I've tried launching it with Open JDK Java 6 Runtime and Oracle Java 7 Runtime, and the same thing happens.
My question is how do I fix it?
I remember once when I first installed Minecraft and Tekkit with this OS, I had to replace some resource files or lwlgl files and it started working, but I can't remember how I did that.