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There's a good post on the O'Reilly Rich Internet Application blog about running Air under Linux on the EeePC:

Adobe recently released the first public alpha version of the AIR runtime for Linux on labs. This is great news! I felt compelled to "geek out" with it, and was able to get AIR running on an Asus EeePC, although with a few minor issues.

The Asus EeePC runs a derivative of Xandros with KDE, which is not a supported Linux distribution for AIR. I got it working with a little help from the Adobe forums, and I'm very excited about it. I have never gotten into Linux desktop application development, but I think that's could soon change.

There's a thread on the Adobe forums that has guidance for running Air on Linux machines. It's a simple matter of downloading the SDK and running your applications from the command line using the adl command like so:

~/AIR-SDK/bin/adl -nodebug ~/app/META-INF/AIR/application.xml ~/app

The AIR runtime for Linux release notes are pretty clear that this is still a pretty alpha product with some unfinished features, but it's something, and if you do a lot of AIR or traditional Flash development, this would be a cool way to include Linux as a build target for your next desktop application.

AIR + Linux + EeePC [via Lebon Bon Lebon]
Adobe AIR for Linux
Running AIR on Linux (Adobe forum)