Hi Will,
I don't have personal experience with such a cleanup job, but I'd first try (on a small spot) a mild detergent/soap and some water on a micro fiber cloth. How to proceed depends on how the base material and the coatings respond to moisture. If nothing comes of, then perhaps adding some washing soda will help to break down the grease film. If nothing helps you might have to try something alcohol related.
But then you probably already contemplated that.
Good luck, and do let us know the outcome.
Cheers,
Bart