Another trick worth trying (I use it after working on bicycle chains, have no idea how it will work for heavy dairy grease) is to use some nice skin compatible oil as a solvent for removing the grease. You can then use your ordinary soap to remove the oil, or you can simply scrape the oil off. Various vegetable oils, 'baby oil', oil based moisturizers, massage oil, etc. are all potentially useable in this fashion.
-Jon
