I was fighting this issue for a few hours, and I found a solution that worked for me. Hopefully it will work for some others as well.
It appears to be an issue with how the launcher uses Java. I couldn't figure out any way to change things using the TechnicLauncher.exe, so I downloaded the TechnicLauncher.jar file. This file is available from the downloads page using the Linux link.
Wonder of wonders, I ran it and it worked.
If it doesn't work for you right off the bat, then at least you can run it using a specific java install and setting other parameters. Best of luck to everyone.