This guide is for making tetrises with a memoryless randomizer. The strategies are biased towards the scoring system in Gameboy Tetris and Tetris Friends -- Tetris 1989 (making tetrises is by far the best way to increase your score quickly), and towards playing in earlier levels where you have more time to think and improvise. Also I'm assuming people would prefer to lose a few games in the early levels while playing aggressively, giving themselves a good chance at setting a personal record score, instead of going for a good average score. Enjoy.
Watching the previewIt's always a good idea to know where you'll put the next piece before it spawns. This becomes more and more important in higher levels in Gameboy Tetris. But if you have time to improvise it helps a lot to watch the preview while the current piece is still falling. It's good to have a general area of the field where you want to put the falling piece, but to consider shifting it over a column from where you had originally planned, or rotating it a different way. It's better to make your stack ugly knowing you'll fix it with the next piece than to try to make it pretty all the time. This idea applies somewhat in all tetris games, but especially when the randomizer is memoryless.
The L piece can fit a few different ways here, so the preview can help you choose which way is best:
O coming next:

S coming next:

I coming next:

More on watching the preview in the skimming section.
Stack shapeYou'll want to maintain a stack that has room for any piece that may come, and at the same time you'll want to have skimming possibilities when you're not getting any I's. The best way I've found to achieve both of these is to build a gradual slope with the highest point at the wall opposite from where you're tetrising:

Obviously you can't keep a perfect slope all the time, but try to place each piece in a way that makes it easy to get back to a nice slope at a later time. It doesn't hurt to build up the leftmost 2 or 3 columns as high as you can. In Gameboy Tetris it's logical to stack high on the left side because I, S and Z rotate toward the left, making it easier to get them to the left wall before they've fallen very far. In Tetris Friends--Tetris 1989 it doesn't matter as much because I, S and Z can rotate toward the left or right, and the lock delay gives you extra time to slide if the piece lands on the stack.
SkimmingSkimming can help you maintain a good stack shape, and if you're giving yourself room to skim you can stack higher while waiting on I pieces. It's a pretty straightforward idea that you want to skim with pieces that don't fit well in your stack, so I'll move on and talk about watching the preview while you skim.
Sometimes you'll know you need to skim but you'll have a couple ways to do it. Often in these situations the preview dictates which way is best. Here's one example:

Now with an L and a J coming, making a skim triple -> skim double is the obvious safe choice when you have a high stack:

But remember in Gameboy Tetris a tetris scores 3 times as much per line as a triple. If you're playing in a level where you can easily handle the speed, there's no reason for a mechanical approach to skimming. It takes a lot of work to build your stack high and you want to keep it there and wait on I pieces as long as possible so you can make the most out of your effort (of course if you can make tetrises while keeping your stack low then do that). As you play more and more you'll get better at visualizing what the stack will look like after you've placed the falling piece, and this skill will help you with planning and improvising.
You probably know you want to skim with the L, so you can go ahead and shoot it over to the right wall after it spawns. Then if you have time and notice the J is coming you might make a quick change of plans:

And now you have a little breathing room at the top, while still having a chance to make a few tetrises. Don't be surprised when 3 I's come soon after each other--be ready

CONTINUED IN THE 2ND POST
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Flexible planningWhile you may be most comfortable building in columns 1-9 or 2-10 and tetrising by the wall, it's good to be flexible and find the easiest way to build without excessive skimming. Your entire game plan may change with each coming piece, especially when your stack is very low.
Here you have a low jagged stack and an L piece coming:

The L fits nicely in both of these places:

But the more you stack up in the center, the more you start depending on an ideal sequence of pieces to fill the sides. This isn't a big problem in games with more predictable randomizers, but when playing with memoryless you just don't know if you'll get the right pieces. Your best bet here would be to fill in the gap on the far right, and plan to tetris in either column 3 or 8:

Notice that now the stack can handle just about any sequence of pieces that may come. Even if you get a string of Z's, you can shift focus and work towards tetrising in column 10:

At this point you would probably want to fill in the left side, clear the third row from the bottom, and continue to build up columns 1-7 evenly until you can tetris in column 10. Eventually after tetrising once or twice and maybe skimming a bit, column 8 will open up and you should be able to tetris there. This is just an example, but the basic idea is to be prepared to tetris in any column when things don't go according to plan.
OverhangsAs a general rule it's a bad idea to make overhangs when playing with a memoryless randomizer, unless the very next piece will fill it. Even if one of several pieces can fill the hole you can't be certain you'll get any one of them soon. There are a few exceptions to this rule though. Here's one example:

While you don't know you'll get T, L or J soon, what makes this situation workable is that there's a fixed empty space to the right of the area you want to fill, and it's preferable anyway to keep that section of your stack lower, which will happen naturally as you stack upwards everywhere else. Also you'll have enough connected space in columns 1-7 to build up evenly while you're waiting on T, L or J.
This is ok but slightly more risky. The plus side is you'll have 8 connected columns for stacking:

This twist setup is fine if you feel confident in your timing in Gameboy Tetris, and it's easy with lock delay in Tetris 1989:


If the gravity is slow enough you can also push a T in from the side (or in Tetris 1989 approach the hole with the T rotated left once, then rotate it right twice after it lands):

Questions, comments and suggestions are appreciated. Thanks for reading.