If they are really loaded with dirt, grime, dried oil, etc, a toothbrush and just a touch of Mineral Spirits on the bristles will clear the checkering nicely.
It's used a lot to clean up old orig checkering on vintage firearms that need just that, but do not need further actual re-cutting of the checkering work done.
Even soap and water can be of use. But that can lead to the grain 'raising' in the checkering and the diamonds becoming a fuzzy field of bloom.
Requiring a complete re-cut to get the diamonds and borders looking like Checkering once again.
Checkering itself leaves a lot of wood area exposed and a lot of different grain. All those sides of diamonds back and forth in the wood..
A prime candidate for the grain fibers to raise,,and water, especially if warm, will do an instant job of it for you even if not intentioned.
Mineral Spirits is safe on just about any decent DRIED and cured finish.
Alcohol with disolve Shellac finish (and so will any cleaners w/ Ammonia in it).
Acetone and any of the super fast grease/oil disolving & evaporating solvents will disolve just about any common finish
Follow-up Scrub the checkering dry with the dried toothbrush to get all of the loosened up crud out.
The mineral spirits will evaporate after a few minutes.
Then use the same brush cleaned and dried to brush in a very thin coating of thinned stock finish, some use straight Linseed Oil (don't over do it or it'll just end up gummed up again as it is kranky about drying).
Min-Spirits is a common thinner for Linseed Oil and oil based stock finishes so any remaining in the wood will just thin these final finish application a tiny bit and then dry. Nearly too small too notice.
You want what ever you use to dry. A thin coating is all you need and just
realize there is nothing that will be completely water proof.
A brushed on & buffed coating of wax over the final product helps. But if you plan on adding additional finish, the wax will resist that finish from linking with that underneath the wax coating.