How to refinish a checker topped table?
You've probably heard the advice that you should sand with the wood grain, and you want to know how to apply that when the grain is going all over the place...  Well, the best bet is to just get a random orbital sander and just sand like a solid piece, ignoring the grain.

If you don't have a random orbital or are sanding by hand, that is fine, but you can't realistically "follow the grain" on a parquet top.  If you are trying to remove a heavy finish, start with a coarse paper (80 or 100) and once the finish or bad spots are gone or mostly gone, move to a finer paper (150 or so).  Depending on the finish and the type of wood the top is made of, you can move to a 220 grit paper for a final pass, but unless you are going for a super-high-quality finish, that is usually not necessary.  If there are noticeable swirls after the 150, then a pass with 220 is warranted, but in my experience that is rare.