Chromium builds for Mac OS X
Currently recommended method (05-2017):

Install HomeBrew
brew tap domt4/chromium
brew cask install mac-chromium
brew tap buo/cask-upgrade
(click here for cask-upgrade info/source)

To upgrade Chromium, run:

brew cu

or

brew cu -y -a

This was the original posted answer:

Install HomeBrew
Install HomeBrew Casks: brew install caskroom/cask/brew-cask &lt;- this is now obsolete
Install Chromium: brew cask install chromium

Or, an alternative
is this method which uses this script, and has a more up to date version as the one above (and also verifies the Chromium checksum), and provides updates:

Install HomeBrew
brew tap domt4/chromium
brew install domt4/chromium/chromium
brew linkapps chromium

updating is as simple as:

brew update; brew upgrade

If you want to automatically upgrade daily in the background you can use this (note that this also upgrades your other HomeBrew packages if you have any):

brew tap domt4/autoupdate
brew autoupdate --start --upgrade