OK, so it seemed like it was easy to install AIR, but after installing one application I couldn't get it to install others.
Turns out I used the right process for 32-bit processors, but I'm using a 64-bit processor. Getting AIR to work with a 64-bit system takes a little more work.
Fortunately, I found some help at OSSRemblings. It's actually step-by-step instructions for installing an AIR app called TweetDeck, but it starts with all the steps to install AIR on 64-bit Linux systems.
I'm planing to install TweetDeck anyway.