Specifically Ubuntu? I've got it installed on my dumbshit nephew's computer, however I have no idea how well it'll run, in comparison to how it ran on his machine when it was Windows 7 (which he infected to high heaven).
Hopefully it's fine, because I'm tired of cleaning up his mess on Windows 7, the OS that's so secure, yet was brought to a screeching halt due to malware.
"java -jar Minecraft.jar"
Ubuntu also should have the option to open it with OpenJDK, if you right click the .jar and go to "Open With" (assuming you have OpenJDK installed).