The flash player available at Adobe download will not work with x86_64 bit browser. Adobe has released the 64 bit player, but it is yet in Labs!. The 64-Bit player beta version can be downloaded from Flash Player 10.1 Beta 64-Bit.
Unzip the download,
tar xvfz libflashplayer-10.0.45.2.linux-x86_64.so.tar.gz
For firefox 3.5.4, just copy the extracted file libflashplayer.so to
/usr/lib64/mozilla/plugins/ folder in Fedora 12.
For Google Chrome, create plugins folder in the directory of chrome installation, which most of cases will be /opt/google. Then copy the libflashplayer.so to plugins directory.
Now start chorme with
google-chrome --enable-plugins
Now flash content should display!