On top of AIR, you can install and run both Tweetdeck and Twhirl. Each has their advantages and I'm not going to address them in detail here. I just want you to know you can get them, you can install them easily (just allow Adobe AIR Installer to open the download, rather than save the file) and you can run them on Linux - 32-bit or 64-bit.
The one caveat I have found so far is that running an AIR app (Tweetdeck or Twhirl) on 64-bit Ubuntu, will cause the Adobe Flash player for 64-bit Linux to crash. The Flash player is only available for 64-bit linux as an alpha version, so bugs are to be expected. YouTube mostly works OK for a while, but if you go to a site like Metacafe, your browser (Firefox, SeaMonkey - whatever) will crash instantly as it tries to load the Flash content on the home page. This problem does not occur on 32-bit Linux. I've opened bug FP-1670 with bugs.adobe.com.
The cropped screen capture in this post is of Twhirl 0.9 running on my 64-bit "Intrepid Ibex" Ubuntu 8.10 desktop.
