My guess would be that mineral spirits or paint thinner would be pretty
good at removing grease.  We've used it for cleaning grease off of
parts on our carousel during and after restoration.  In wood, it will
probably take several to many applications to thin and draw the grease
out.  Stains may remain, but they could then be bleached or pickled as
necessary.

If you've never used odorless paint thinner you might give it a try.
It sure is nice not to breathe those fumes, especially in tight places
indoors.

Tommy Forney
Shelby NC