I have 3 objects. I select all, and would like to rotate each around the Z axis of each object (locally) not global.
I need to rotate each object about its local axis.
First set the Pivot Point to "Individual Origins".
Now change the Transformation Orientation to "Local".
Both settings are found in the 3D view header.
Now when you rotate, either by dragging the blue handle of the rotation manipulator, or by pressing R ZZ it will work like you want.
There's a button to pick transformation mode, just above the timeline in default view. It will be showing two orbs overlapping with a blue mark in the middle.
Click that and select the "Individual origins" setting (two orbs with individual blue marks).
Now every object will rotate around its own origin point.