Recently I've upgraded both my laptop and desktop to 64 bit. I wanted to share how I got Fedora 14 working on my laptop and Red Hat Enterprise Linux 6 working on my desktop with Adobe Flash Player 10.1.
On both distributions, I found the same solution for how to get Adobe Flash Player worked. It was not installing an rpm file or setting up a yum repositority. Instead, I downloaded http://download.macromedia.com/pub/labs/flashplayer10/flashplayer10_2_p3_64bit_linux_111710.tar.gz from the http://labs.adobe.com/downloads/flashplayer10_square.html page. Adobe's download links lead me in circles when I first tried to get this working, but I finally searched their site and found this page to download the plugin.
To get Firefox to work, I extracted the downloaded file libflashplayer.so to each user's home: ~/.mozilla/plugins directory. For Google Chrome 64 bit to work with Adobe Flash, I had to create a /opt/google/plugins directory and then copy the extracted libflashplayer.so file into it. Remember to close all browser windows, and restart the browser for the plugin to take effect.
I resisted going to 64 bit Linux up until recently for this very reason: support for getting common things to work in 64 bit Linux lags, at least as of the end of January 2011.