Dan you are right on.
Peter, most spatial analyst tools only operate on one band at a time, if you pass one of these tools a multi-band raster it will simply pick the first band and process that. The best approach if you need a mean R, G, and B value would be to run Zonal Statistics As Table on each band in turn, and combine the results using the Join FIeld tool.
Cell Statistics gives you the average value pixel by pixel through a stack of bands, I don't think that's what you want.
What is the purpose of your mean value? Maybe what you really want is the Create Signatures tool, which does full statistics for each of your zones that can be analyzed using some of the other tools in the Multivariate toolbox.