I have had my nose to the proverbial grindstone these last few days, working hard to get a Blender animation to play inside of Unity. After two solid days of coding, debugging, and scouring the Web for clues, I finally got it to work:
To summarize my findings: Unity doesn't automatically import mesh animations from Blender. Rather, what you need to do is to create an armature, make the mesh child object to the parent armature, and then animate the armature. The animation needs to be saved as an action in the Blender Action Editor, which Unity picks up once you import the file. Work on the animation spanned three development platforms, Blender, Unity and Microsoft Visual C# Express (which I configured for automatic code completion). Now that I have a rough idea of how to get objects to move within the game, I'll add some extras such as sound and screen-based text. Should be a lot of fun.