Flash Programming on Linux
You may find some useful suggestions over here.
I use Flex Builder for Linux to build applications using Flex. However it can be a bit fiddly to set up currently due to it requiring newer the latest version of the AIR SDK (see the release notes on how to upgrade). Also the MXML editor broken with Eclipse 3.4, so you'll want to stick with 3.3 until they release a patch, or patch 3.4 yourself according to this (untried by me).
In addition to Flex Builder, you can use FDT by PowerFlasher, awesome product.
Website:http://fdt.powerflasher.com
I don't mean to sound a bit mean, but Noldorin's answer is no good at all and misleading, in fact it is fact-less. Flash development restricted to Windows? REALLY?!?!