This is one of my favorite parts of the process. Use a quality 2-part cleaner/bleach such as Te-Ka or Snappy Teak-Nu. I've tried many and these are the only ones I've found that do a reasonable job. Store brands do little or nothing, and all of these treatments are not of the same chemical formula. Instructions say not to use on mahogany, but in my experience on many boats, they pose no problems for mahogany, and in fact clean it beautifully while leaving it and all other wood with an even beautiful patina. The instructions on many also say "scrub-less" or "no need to scrub". Not true- use a soft nylon brush to scrub the treated surface if you really want to clean it.
Are these things too harsh? One British book says never use them, but in fact, they pose no real harm (again, in my experience) to even soft woods like cedar. Yes, the surface becomes fuzzed a but, but sanding with 120 followed by 220 makes a beautiful surface for varnish.
Finally, oxalic acid- wonderful for iron stains. If you had any iron fasteners, use it to remove black iron stains. It also works on black stains in tannin-laden woods like white oak. Make a concentrated (saturated) solution, and paint it on. You'll see iron stains disappear before your eyes. I've heard all sorts of ideas on neutralization of the acid, but use only water- the oxalic acid is readily soluble in water, so it will wash away. I've even read that you should neutralize with vinegar (acetic acid will not neutralize oxalic acid!), or that you should use a base like sodium hydroxide. A base like NaOH will neutralize an acid, but it can also turn wood black!
Michael