Get Raster Values from a Polygon Overlay in ArcGIS

I have the same problem as Curlew in Get Raster Values from a Polygon Overlay in Opensource GIS Solutions and thecrashlandingdodo in Step-by-step: How do I extract Raster values from Polygon overlay with Q-GIS or R?. Curlew says it's rather easily done in ArcGIS, but I just don't know how and R is on my to-learn list.

I have many polygons in a shapefile and a raster with different values. I would like to know for each polygon how many percent each raster value has in that polygon. Any of the zonal stats does not seem to do this (or did I miss something?).

Any help would be appreciated.