I would go the mineral spirits route. Have used it many times on old varnish and shellac piano finishes that are dark to black with age. It is very gentle and will also remove any built-up waxes and polishes that may have been used. Also, doesn't stink nearly so much as turpentine. I would shy away from the turp - it is a much stronger solvent and will dissolve many finishes. A toothbrush would work well in the checkering. If you want a bit more aggressive cleaner on the straight wood, mix just a bit of pumice powder in with the mineral spirits. Pumice is used to polish shellac, lacquer, and varnish finishes.
