Chromium 34 in the main repos have started using Aura (early), which does not include support for NPAPI (this is a planned phaseout of NPAPI in Chromium). Therefore, you need to use Pepper Flash to be able to use Flash.
Ubuntu 14.04 (Trusty) and newer
If you have Trusty, you can just run
sudo apt-get install pepperflashplugin-nonfree.
Ubuntu 12.04 (Precise) and newer
If you don't have Trusty, you can use this PPA to install Pepper Flash for any supported Ubuntu version above Precise. Run the following commands to add the PPA and install Pepper Flash:
sudo apt-add-repository ppa:skunk/pepper-flash
sudo apt-get update
sudo apt-get install pepflashplugin-installer
Note that you need to configure Chromium to use Pepper Flash. To do this, open
/etc/chromium-browser/default and add the following line to the end of the file on a new line:
Close all windows and re-open.
Updating Pepper Flash (on Trusty)
You can run
sudo update-pepperflashplugin-nonfree --status to see what version of Pepper Flash you have installed. If there is a newer version available, you can just run
sudo update-pepperflashplugin-nonfree --install.