I know how to do a 3-cycle, I was simply asking if there was an algorithm that was faster
Sure there is but there is also "Newton's law of twisty puzzles" which states "for every rotation of a piece there is an equal and opposite rotation elsewhere". This holds true for all commutators and also whenever twist is preserved for the piece type which is the case for most puzzle's pieces including the 3x3x3 / 4x4x4 centers. Little known fact: Sir Isaac Newton was a twisty puzzle genius.
If you want to rotate a group of 4 centers clockwise and another group of 4 centers on a different face counter-clockwise then you can use what I have been terming an "orientation changing commutator" which using Gelatinbrain's 3x3x3 notation looks like:
[F&2, U'&2, F'&2], U, [F&2, U&2, F'&2], U'
From your original picture this would help you a great deal. It would solve all 4 centers on one face and solve 1 center on the other face leaving you with a single 3-cycle (or possibly 2-2 swap) rather than a bunch of 3-cycles.