I tinkered with Gimp for a bit, and after coming to the realization that trying to animate layers with transparency so that underlying content is visible would be impossible, I decided on a gravel texture to animate behing the spinning logo. I thought maybe I would draw a kind of harness attached to the Ubuntu logo, and connecting to something differently in the banner. I'll keep the intersecting lines stable, so that even if the 2 animations lose sync, the 2 lines will connect.
I've set this as my Google+ logo, but I'm going to release it under some free license.
If you like the animation, please let me know below in the comments, and if you want to know how I did it, I might do a tutorial if there's demand.