Apparently this is a known bug since 4/4/12.
If this is happening to you, go to the bug and mark it as affecting you.
I fixed this problem (by stumbling around).
Since I have other, older distributions working with Compiz, I opened ccsm from one of them and compared them to the 12.04 upgrade.
Strangely, after enabling Windows decorations, move, and re·size the problems stated is this question were solved.
By setting others from the older ccsm, I got a rotating cube.
Alas, there are no more 3D windows
UPDATE: 3D windows are available for Gnome Classic. The Compiz PluginsExtra is required. Synaptic is required to install Compiz PluginsExtra, it is not available in Ubuntu Software Center. And they work very nicely!
Screen shots of the older (10.10), working ccsm window can be seen at https://bugs.launchpad.net/ubuntu/+source/update-manager/+bug/994893
UPDATE Although not mentioned in release notes, cube and 3D windows are not a part of Ubuntu. Compiz settings may changed when logging out of Gnome Classic and into Unity, then back.