Path the .flv relative to the WEB PAGE that the .swf will be on, NOT the physical location of the .swf.
Almost always when it works on the local machine and not the server, it's a pathing problem.
You can put your Flash related files in whatever folders you want, they do NOT have to be in the root, they do NOT all have to be in the same folder. But if you have a problem and if sticking them all in the root folder works, then you know that the issue was a pathing problem.
Just remember that paths used in the .swf become relative to the Web page on which the .swf is placed, NOT it’s physical location. So for example, if your .swf is in the flash/data folder and you use that .swf on a Web page in the root folder, you are in effect, removing that .swf from flash/data and putting it in root. So if the .swf is loading any related files (xml, images, video, etc), the path used inside the .swf to load the .xml file has to be relative to it's new location in root and then back down into flash/data. This is true even though when testing the .swf by itself, it can be inside flash/data and work just fine, since relative to it's location, the path is just fine, they are in the same folder. But if that same path is used when the .swf is placed on a page two folder levels up, the relative path has changed, the old "same folder" path will not work.
In fact if you are placing the .swf on a web page in a different folder than the .swf is stored in, and that .swf calls external assets, then direct clicking and opening of the .swf in it’s folder should NOT work! That’s because the paths to the external assets should be relative to the Web page and not the physical location of the .swf.
So just be sure that you use addresses relative to the final Web page locations (not physical file locations) and you can put the Flash related files in what ever folders you want.
Eye for Video