Linux users must compile Cube Trains from the source at the moment. You can read the full summary of this download for more info.
From the developer, David D. Roberts:
Linux users must compile Cube Trains from the source, at the moment. There is an old compiling guide which may still work, or at least provide a good starting point. Sorry for the inconvenience.
In addition, it has been discovered that under most linux distros, the editor's "copy to clipboard" feature is miserably broken. Unfortunantly, I am unable to find a fix for this. The workaround for this is to:
1) Run the game from the console.
2) Copy level to clipboard, like normal.
3) Press ctrl-d to bring up the debug console, ctrl-v to paste the level into it, and then press the enter key to evaluate it. (Has the side-effect of printing it to console.)
4) Copy the level from the console window. Note that you will now have to restart the game before pasting anything.