The Mint 14/15 LiveCD worked fine on my old Compaq 6820s but after installation Mint would hang with a black screen during the boot process. After much rebooting and editing the grub config (hold down shift key during BIOS POST process to display the Grub menu) I found the solution that apparently everyone knows. Until I knew the answer I didn't know the correct question to pose to Google.
For some reason the Radeon video card doesn't like the graphical splash screen and has to be turned off with "nomodeset" command-line option, something like this:
sudo pluma /etc/default/grub
Change this line:
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash nomodeset"
And apply the change with:
sudo update-grub
