I've just upgraded my computer to a new quad-core AMD64 system and did a fresh install of Ubuntu Karmic 64 bit. My old tutorial on getting Tweetdeck working still mostly applies.
My old dual-core AMD64 that was about 3 years old worked just fine with Adobe AIR, but with a newer AMD 64 CPU, Adobe AIR will segmentation fault when you try to run the application installer. This means you can install AIR just fine, but you can't install any programs that actually USE it.
I've tested versions of AIR including 1.5, 1.5.1 and the latest 1.5.2 and all three segfault running the installer.
HOWEVER - the Beta 2 of AIR works just fine with it.
Now, if you previously had TweetDeck working, you might need to uninstall anything AIR, including AIR, then delete the ~/.appdata directory. Afterwards, do a fresh install of Air followed by Tweetdeck. I had originally copied my home directory from my old computer and then tried to reinstall Tweetdeck and Tweetdeck initially complained that it wouldn't run on my computer.