Use a good stripper like Formby's to remove the old finish.
Use a solvent-proof toothbrush or other small brush to scrub the finish out of the checkering.
Let the stripper do the work.
IMPORTANT! After stripping the finish use something like naphtha or acetone to wash the piece down. this will neutralize any stripper left in the wood pores. If you don't your million dollar finish will start to show little gas bubbles after a month or two.