What to do when a Linux desktop freezes?

If all else fails, you Raise The Elephant.  Essentially, there are special Magic SysRq key sequences (Alt+SysRq+?) that the Linux kernel handles specially.

If your Linux box freezes and simply won't yield to any other key-commands, you should definitely try one particular key sequence before a hard reboot.

The key sequence is popularly remembered with the mnemonic:

Raising Elephants Is So Utterly Boring

• Alt+SysRq+R switch keyboard to 'raw' mode
• Alt+SysRq+E send SIGTERM (termination) signal to all processes except mother init
• Alt+SysRq+I send SIGKILL signal to all processes, a little more aggressive
• Alt+SysRq+S sync all filesystems to prevent data loss
• Alt+SysRq+U remount filesystems as read-only
• Alt+SysRq+B forcefully reboot

You can try Ctrl+Alt+* to kill the front process (Screen locking programs on Xorg 1.11) or Ctrl+Alt+F1 to open a terminal, launch a command like ps, top, or htop to see running processes and launch kill on not responding process.

Note: if not installed, install htop with sudo apt-get install htop.

Also, once done in your Ctrl+Alt+F1 virtual console, return to the desktop with Ctrl+Alt+F7.

In most distros pressing Ctrl+Alt+Backspace kills the X11 (graphic) interface and restarts it. Unfortunately some recent, supposedly "user friendly" distros deactivated this very useful shortcut for some unfathomable reason. I don't know if Mint is so "user friendly" but you have nothing to lose trying it :)