I always had problems getting Flash to work in both Firefox and Chrome when running Fedora Linux. This became more of a problem since I started using the x86_64 kernel. But here are the steps I took that worked under Fedora 16. Thanks to the_nins on the chrome support pages.
Install the flash rpm from http://get.adobe.com/flashplayer/
Install the other required packages
yum install flash-plugin nspluginwrapper.x86_64 nspluginwrapper.i686 alsa-plugins-pulseaudio.i686 libcurl.i686
Create the wrapped up plugins for x86_64
mozilla-plugin-config -i -g -v
Create a symlink to the wrapped file:
cd /usr/lib64/mozilla/plugins ln -s /usr/lib64/mozilla/plugins-wrapped/nswrapper_32_64.libflashplayer.so
Restart the browser and it should pickup the flash plugin. I would also highly recommend using Flashblock for Firefox to load flash animations on demand.