When I download flowplayer locally on my machine and open index.html, the example video runs fine, no problems.
I have a linode running cherokee 1.2.2 with a basic 'List and Send' virtual server setup. If I simply download the flowplayer zip, unzip it in my virtual server's root directory, then navigate to the example directory, I see the page fine but the flash video section is just black, with no video and no controls.
I'm not changing a single file in the flowplayer zip, just extracting it. Looking at my cherokee access log I can see that the swf file is being served fine.