The good news is that the Compiz cube settings survived the upgrade to Ubuntu Raring 13.04. When all was said and done, the cube still worked great.
The bad news is that although the Compiz cube worked fine, the setting necessary to make it work broke the new Workspace indicator on the Unity launcher. The cube requires that the workspaces be arranges as a 1 x 4 – all in one row. The Unity workspace indicator requires that the workspaces be arranged in the default 2 x 2 square. This makes the cube a flat plane because it only uses the top two workspaces and also only switches between those two.
So, I used Compiz Configuration Settings Manager (CCSM) to unselect all the Desktop Cube eye candy and enabled the Desktop Wall:
FYI, the Desktop Wall was the default setting in Compiz when I first installed it.
Next, I ensured the default 2 x 2 workspace configuration by deselecting then reselecting Enable Workspaces under System Settings -> Appearance -> Behavior:
The net result is that I now have the 2 x 2 arrangement on the Desktop Wall:
While it’s not as sexy as the Desktop Cube, I can still switch between workspaces with the scroll button on my mouse. But more to the point, the indicator (last icon on the Unity launcher in the screenshot) correctly shows the presently active workspace (workspace 1 in the screenshot).
As best I can tell, the wall only uses window sliding when changing workspaces. The settings don’t seem to offer any other transition like flipping.