Paper towels or rags to wipe up 80% of the old grease should
work fine. Being compulsive, I'd follow that up with paper towels
or rags soaked in some solvent to remove about half of what
the first pass didn't get. The extra effort isn't that excessive
considering the fact that you are only doing this every 10 years
or so.

Use something safe and inexpensive as the solvent. Kerosene
(fuel oil) would seem like a very good choice. Brake cleaner
works extremely well, but it is both expensive and the fumes
are very unhealthy.