I understand that the Linux builds of Chromium are still very alpha, but I'd like to enable the flash capabilities.
How do I get flash working in Chromium?
First of all, copy the flash plugin.
Then start Chromium like this.
This answer is outdated; you no longer need to do this to get flash running in Chromium.
Chrome supports flash (and plugins generally) in recent builds. It was featured on lifehacker not so long ago with links to repositories.
They are experiencing some technical diffuculties, the site is slow and more down than up. I am happy that I managed to copy the url from the search results.