While searching for something else, I stumbled upon this list of fifty open-source Linux applications (and this follow-up list of fifty more). Many of them I already use, or at least knew about. But there were some which I hadn't heard of before.
Such as Synfig Studio, a 2D animation package which is capable of producing works like this. It was already in the Debian package repository, so it was a simple matter of sudo apt-get install synfig synfigstudio to get it up and running. I've only toyed around with it a little, but its toolset leaves Flash's in the dust. (At least as far as non-interactive animation is concerned.)