Hi reader,
Due to the economy and lack of interest ($), Adobe seems to have largely abandoned Linux platform support for Flex/Flash Builder.
If you're running a 32-bit Linux build or have the patience to maintain a 32-bit stack on your 64-bit OS, there's the aging Flex Builder 3 for Linux Alpha. As of this post, the most recent drop is 24-Nov-2009, but don't be fooled by this - the only updates in the last couple years have been security fixes.
I currently run an XP WMWare Server Virutal Machine for my copy of Flash Builder 4, but the performance hit is tough to swallow if this is your primary development environment. One option here is to edit in the VM on a SAMBA/nfs networked project/workspace that is build with maven, etc. on Linux.
One bright spot on the near-horizon is the AXDT Eclipse Plugin developed by Martin Schnabel. I've used his current (soon to be previous) version to provide ActionScript syntax highlighting and some building, but Martin's been very busy working on a reimplementation that shows a lot of promise. I encourage you to visit axdt.org/new.axdt.org and support his work.
In the meantime, I've just become a Flash Builder "advocate" here at CodeWeavers, and will see if the forums and experts that gives me access to can get us any further to running FB4 on Linux 64.
There are a couple other $ and free options out there that I haven't really tried - if you have experience with any of these, please add a post to this forum, or at least subscribe and I'll keep you posted on what I encounter in the future...
Chris Ingham