I would like to animate a mesh's geometry without using shape keys. Is this possible?
Yes, you can do this very easily by using the AnimAll addon. "AnimAll" stands for "Animate All" because it enables you to animate many things that cannot normally be animated in Blender.
Here's how to use it:
NOTE: The tutorial video on this page describes how to use all the many other powerful features of AnimAll.
Animation in blender, there are lots of options.
There are more ways to animate meshes in blender, but this is just a sample. It should be noted that if you don't mind animating in edit mode, that the AnimAll addon can be used to save a step in place of the hooks in the above methods. If you would like more detail on any of these methods leave a comment.