I use a toothbrush on the checkering but don't use it on the smooth finished wood, it'll scratch the finish. the linseed oil (or other cleaner/softener) will soften the old oil & dirt in the checkering, allowing the toothbrush bristles to loosen and remove it. Purple Power (brand name) has also worked well for me.
Regards, Joe

You can lead a man to logic but you can't make him think. NRA Life since 1976. God bless America!