I decided to keep it simple; just a text editor with command-line compiling and testing.
So here is how I did it:
1. Install the flex sdk:
The free SDK is downloadable from adobe ( http://opensource.adobe.com/wiki/display/flexsdk/Download+Flex+4 ), I downloaded 4.5 to try out some of the new features, but you can choose the build that meets your requirements. Save the library somewhere to your local disk (I chose /home/USER/developer/sdks/flex_sdk). The next step is to enable the terminal to access the executables in the flex_sdk which will build your programs.
Open a new terminal and type sudo nano ~/.bashrc (you may be asked to enter your password) and add the line:
export PATH=/home/USER/developer/sdks/flex_sdk/bin:$PATH
to your bashrc file. To test it has worked restart the terminal type "which mxmlc", it should echo the path you just provided.
2. Download the Flash 10.1 Debugger for Linux:
Download the Debugger and install it using:
$ cd /tmp
$ wget http://fpdownload.macromedia.com/get/flashplayer/current/install_flash_player_10_linux.tar.gz
$ tar -zxvf install_flash_player_10_linux.tar.gz
$ cd install_flash_player_10_linux
$ ./flashplayer-installer
in your terminal.
Hopefully you'll have a functioning debugger now. Check it works by finding an swf, setting all swfs to open with the flash player, and run gnome-open file.swf, and the debugger should open in the flash player.
Failing this just use the flash for Firefox plugin.
3. Install AS3 and MXML language files for Gedit:
If you don't have Gedit, get it, its a great little text editor, and for me at least it does everything I need to (at least for editing AS3 files). Download the MXML and AS3 files from http://conic.se/blog/posts/8/ and copy them into ~/.local/share/gtksourceview-2.0/language-specs/. When you run Gedit you should be able to switch to the correct syntax highlighting for each language.
Finally put all three together. Create a directory containing the actionscript library you want to compile, and in your command line run mxmlc test.as. If you've followed the steps you should have created an swf. Finally run the swf using gnome-open test.swf.
Here are a few screenshots of the system in action:
Good luck developing AS3 in Linux