I recently installed Fedora 12 and was at a loss to get Flash working under Chrome. Here is how I got it to work:
1) Install Google Chrome via the RPM installer.
2) Install the flash plugin and dependencies:
yum install flash-plugin
yum install nspluginwrapper.x86_64
yum install nspluginwrapper.i686
yum install alsa-plugins-pulseaudio.i686
yum install libcurl.i686
(they can be all on one line, but I like to see each one)
(some might already be installed, just move along)
3) Make a link to the plugin for chrome: (as "su")
ln -s /usr/lib64/mozilla/plugins-wrapped/nswrapper_32_64.libflashplayer.so /opt/google/chrome/plugins
4) Change the Chrome menu item to start with "--enable-plugins"
5) Start Chrome and go to about:plugins if you see the flash player in the list then you are done..