Try (on a small patch) Neat household bleach, Oxygen bleach, or I would always try Oxalic Acid or one of the proprietry teak brighteners. Try writing to one of these manufacturers.
I personally would go for Oxalic Acid
Black stains are the first sign that wet rot is starting to attack the wood. Teak is very resistant to rot, so usually its quite curable, but you must get the wood protected, particularly at end-grain.
I've not found any products that totally remove these marks, but dilute oxalyic or phosphoric acid, or the proprietory teak-cleaners made out of them, help quite a bit. Finish off by hand-sanding back with fine wet-and-dry to get an even colour again. If you aren't going to paint/varnish, 600 grade will leave you with an almost polished surface that is hard for rot to penetrate.
If the black stain runs deep you should take steps to dry the wood out really thoroughly, and treat it with a fungicide.
I have a 20 year old boat that was badly neglected in the past and had many stained areas. I sanded most to remove them or covered them with new trim. However I have heard that oxalic acid will bleach it out. Never tried it but you might look into it.