Had an issue where YouTube's HTML5 video wasn't working in Chromium on Ubuntu 12.04 LTS, and didn't really want to install Flash Player if I could avoid it. (HTML5 in Firefox was working fine.) All I got in Chromium was a black screen with a spinner going endlessly,.
In the end, the solution was as simple as missing codecs:
sudo apt-get install chromium-codecs-ffmpeg-extra
It'll uninstall the currently installed chromium-codecs package and replace it with this one. You might also have to restart Chromium if it's running; it's a couple days back since I did this, so can't remember.