I'm not aware of any linux software that deals with flash, certainly nothing that will open .fla's. There are some attempts to make a linux port of Flashdevelop (a free Windows/.NET based IDE for the Flex 3 SDK), but I don't think they've got very far with it yet.
As Vishesh has already said, you could try using Wine to run one of the flash IDE's. Flash mx2004 will probably work well under wine, flash 8 might also work, but I have serious doubts that CS3 or CS4 will work properly.
The windows version of Flashdevelop might also run under wine.
Another usually overlooked alternative is the Adobe Flex3 SDK, which can be used to create flash content in Linux using pure AS3 or AS3 and mxml.
The flex3 SDK ships with some multi-platform command line tools, the bin folder of the SDK contains binaries for windows, mac and linux. To get it, all you need to do is download it from Adobe and unzip it somewhere on your system. And that's it, you're ready to start developing flash in Linux!
The only pre-requisites are that you need to have java installed on your Linux system...But that's not really a stretch, if your machine doesn't have Java, open up your package manager and download and install it from your linux distros repository.
To use the SDK in linux, you simply use a text editor like gEdit, vi or emacs to create your mxml and/or actionscript 3 files and then use the 'mxmlc' command-line tool in the bin folder of the SDK to compile your .swfs.
Here's an example of compiling a .swf in Linux:
Picture this if you will....
1. You've used your favourite text editor and created an actionscript file called
myFirstFlex.as and saved it in in a folder at
/home/your_name/Documents/AS3code
2. You also have the flex 3 SDK unzipped to a folder at
/home/your_name/flex3SDK.
To compile the actionscript file into a .swf you'd need to open up a shell and enter the following:
/home/your_name/flex3SDK/bin/mxmlc /home/your_name/Documents/AS3code/myFirstFlex.as
This runs mxmlc (the adobe AS3 compiler), passing your .as file as a parameter. mxmlc will then attempt to parse and compile your actionscript file into a .swf.
If any errors are found in your actionscript, mxmlc outputs some very handy error messages. So all you need to do is reopen your AS file in your text editor, fix the problems in your code and run mxmlc again.
If (or when) there are no errors, mxmlc will output a .swf with the same name as the input file, so in this example it would output myFirstFlex.swf.
And that's about it. It's not particularly complicated!
There are a few people out there on the web who have created plugins for gEdit and various other linux text editors which allow syntax highlighting of AS3 and MXML. So if you'd like some syntax highlighting in your favourite text editor, try doing a search (e.g. 'gEdit AS3 syntax highlight') and you should be able to find something!
Since I've been using my Linux PC more and more and seeing as I seem to be messing with flash rather a lot, I have been meaning to make a start on a wxWidgets based IDE for the Flex 3 SDK using either C++ or Python. But I've got a feeling it's going to be on the back-burner for a little while as I have much bigger fish to fry at the moment!
Anyways, hope this has helped.
Cheers for now,
Jas.