Even more weird behaviour. The Compiz window manager that comes with Fedora Core 6 uses DRI 3D acceleration for rendering the desktop, and this seems to cause problems with IntelliJ. IDEA starts up fine, but the main window is completely blank - no menu items, nothing. Key shortcuts still work though and popups are visible.
I suspect this is something to do with Swing and Compiz rather than IntelliJ and Compiz, since Eclipse (and other SWT based programs such as Azureus) works fine.