I know a bunch of guys who have been following the following procedure for years and swear by it.
Here's what you need to buy.
1) A bottle each of Sudsy Ammonia and Clorox. If you're going to do decks, buy a gallon each. Combing boards, a quart of each should do it.
2) You'll need a common boat scrubbing brush and a package of fine bronze wool for awkward unreachable spots. I prefer the dry crystal Tip-Top Teak Cleaner, Tip-Top Liquid brightener, and Tip Top Teak oil.
Start by just plain wetting your teak. Pour about a quart of sudsy ammonia into a bucket and fill the bucket to about 3/4 full. If sudsy ammonia is not available, a shot of Dawn Dishwasher liquid and plain ammonia will work.
Dip in your scrub brush and slather the area well with the solution. Keep doing that so it sinks in and start scrubbing in the direction of the grain. Slowly but surely you will start to see the sudsy areas getting dirtier and dirtier looking as the dirt, pollen, fish blood and assorted molds start to leave your teak.
Rinse off with fresh water and do again until you get the sense that your teak looks a lot better than when you started.
Now take a cup or two of Clorox and mix it with fresh water in a clean bucket. Brush it all over your teak and let it dry. It will dry pretty quickly. Some of the black streaks you had were probably mold and this should help clean them up.
If you've never worked with the crystal Tip-Top Teak cleaner before, it looks like a plastic bottle filled with Kosher Salt. Throw 2-3 hand fulls into a bucket and gently fill to about 2/3- 3/4 full and stir so the crystals dissolve. If you're doing a deck, I suggest working this aspect of the job 1/3 of the deck at a time until the whole deck is crystal-cleaned.
Wet the teak with your hose. Then lightly sprinkle the wet surfaces you plan to clean with some crystals you've taken out of the jar with your hand. As they fall on the wet teak, you'll notice red dots forming around each crystal. Don't panic. Take your scrub brush, dip it into your crystal/water mixture bucket and start scrubbing the area you sprinkled. Again in the direction of the grain. You will see two things happen. MORE dirt will appear coming out from your teak, and your teak will start to take on a deep reddish color. Again-don't panic. Clean and rinse with fresh water and keep all areas wet as you repeat this action on each panel of your deck. Use your bronze wool pad in areas your scrub brush won't reach. Once all your teak has been well scrubbed and taken on a reddish hue, you're ready for the next step. Keep it wet as you prepare.
Take your Teak brightener (a much smaller bottle...and it's liquid) and pour about an inch or so into a smallish container and then fill the container 3/4 full with fresh water. Take a throw-away rag, wad it up and dip it into the clear mixture, then get on your hands and knees (if you're doing a deck) and just surface wet an area of the already wet teak. Amazingly, you will see in about a minute or less, that the area you have just done has changed from deep red to golden teak. Do all the wet surfaces. What you haven't done will stay dark red so it's not easy to screw up. What splashes and leaves spots will even out by just wiping over them. After you've done the whole area and have stood up to look at your handiwork, rinse it down again with fresh water.
Now-how it looks now (wet) is how it will look after oiling. How it looks when it dries is another look which many people like. Your choice. Don't oil until it's nice and dry. A few hours in the sun will do it.
Good luck.. Keeping teak clean is a lot of work.