Black spots propagated by water are usually a chemical reaction between iron and natural tannins in the wood. An oxalic acid bleach is what you need to remove them. Be careful with it as the solution and dust are irritating.

http://www.woodcentral.com/bparticles/woodbleach.shtml

Google remove water stains from wood with an iron, works like a charm, not sure since you have sanded but may be worth a try

Using an iron is to remove white blush (entrained moisture) from the finish. That is not the case for the OP, as the finish is removed and the black stain is in the wood.

I've had varying degrees of success with oxalic acid treatment. If it doesn't restore the appearance to an acceptable level for you, you could sand with a coarser grit to take the surface down faster, or plane the surface (as long as it's not end grain).

kudz and uncle Bob make good points. I'd try Bob's first and if you must resort to sanding,I have a suggestion that "might" help. Sanding the stain out will be simple,keeping top level is what require's all the labor. Depending on size of top,how it's constructed and how it detach's,hireing a cabinet shop to run it through a belt/drum sander will save a lot of labor and give it a level surface. Another option is using scrapers instead of sanding to control dips and uneven surface.