There is now finally an official upgrade mechanism for Homebrew Cask (see Issue 3396 for the implementation)! To use it, simply run this command:

brew upgrade --cask

However this will not update casks that do not have versioning information (version :latest) or applications that have a built-in upgrade mechanism (auto_updates true). To reinstall these casks (and consequently upgrade them if upgrades are available), run the upgrade command with the --greedy flag like this:

brew upgrade --cask --greedy

To get outdated:

brew outdated --cask --greedy --verbose