I've done all my Flash game dev in Linux, on my old underpowered laptop. The expensive Adobe CS doesn't have any Linux support, but this doesn't matter since I'd rather write ActionScript 3 code directly anyway. The bigger problem is that the free and open source FlashDevelop IDE is only available on Windows, and most of the documentation for making Flash games outside the Adobe Creative Suite revolves around FlashDevelop. Fortunately you can just use the command line to compile your code with the Flex SDK and bypass all those big bloated programs entirely. This gets you a workflow that not only runs well on older computers (like my cruddy laptop) but is also cross-platform. I used this tutorial from HobbyGameDev to get started:
$0 Flash Development: Quick Intro to MXMLC - This is the first part and explains how to setup the Flex SDK and how to use the mxmlc compiler.
Actionscript 3: Flash Game Core Tutorial - This cover the basics of the AS3 language. It's what I first used to learn the fundamentals of Actionscript.
Actionscript 3: Game Motion Tutorial - This tutorial covers smooth sprite motion.
This post by Draknek covers setting up and using the Flex SDK on Linux specifically: AS3 Flash Dev in Linux