Try flushing with water first, then soap and water, and finally alchol to remove the grease. The trouble is the heavy grease is probably for taking out play as much as for lubrication. Light machine oil probably will leave too much slop in the fit of the gears so petroleum jelly might be your best, if sloppy, substitute. You'll have to decide if periodic cleaning and relubing is worth the hassle because petrolium jelly will trap dirt quickly.