Ubuntu and Multiple Monitors – AMD Edition

First paragraph summary:
There are several ways to end up with a satisfactory experience on the desktop with Ubuntu despite their recent confusion of the user interface. We will discuss some of those another day (KDE vs. Gnome vs. Cinammon vs. Unity). Today we are going to talk about setting up your desktop environment for multiple monitors. This article assumes you are running Ubuntu 12.04 LTS or 12.10, however, the process should work equally well back to version 10.04 LTS unless otherwise noted.