Depends on the finish. If it is any type of natural finish, paint and varnish remover will soften it enough to allow it to be wiped off. Use an old toothbrush in the checkering after applying the remover. It will remove the finish ithout damaging the checkering.
OTH, if it is a later synthetic finish, or an epoxy, mechanical removal (sanding, scraping) might be the only way to remove the existing finish. Best suggestion is to try different removers/solvents and see if any of them soften the finish.