The solution is simple.
Step 1: Get Google Flash Plugin for Linux. See reference to other post at the end.
Step 2: Run chromium with command line switch "--ppapi-flash-path"
Example:
$HOME/chrome-linux/chrome --ppapi-flash-path='$HOME/chrome-linux/libflashplayer.so'
I borrowed that idea from this post, so all credits should go there. You can find more details there as well.
-= Oleg =-