There are two schools of thought on teak maintenance
. Some like the brown look and others like the grey look. I prefer grey so I'll be telling you how to keep the grey.
Start with a good scrub with hot water
and soap and a stiff bristle brush then rinse clean. See what that does to the marks because it might be enough.
If more is needed then grab a cup of white vinegar and a scotch brite pad and scrub not just the stains but the entire area around them. You are doing this to get an even color rather that a funny
looking bleached section where the stains were. Rinse, dry and inspect. It might be necessary to repeat the vinegar scrub if the stains are really bad.
The teak should stay grey with just one vinegar scrub but if you need two then you can expect a color change. No matter what you do there will be less damage than a teak cleaner will cause. That stuff is too aggressive and if used repeatedly will cut the life of your teak in half. Teak cleaner is usually an oxalic acid mix which literally melts away the top layer of your teak. Vinegar is a cheaper and milder alternative.
If you like the brown look of teak then go ahead and use the commercial
products. If your teak lasts 10 years you will have done well.