What do players do when they can't find a move in the middlegame?

In some of my games I can't find a single good move in the middlegame. I can't find a way to threaten my opponent, I don't see any threats I need to deal with, and if I move any of my pieces or pawns, I think I will just be placing them on worse squares. Once I even moved my king to h8 and back to g8 on consecutive moves just because I couldn't think of anything better. What do players do in such a situation?

Tactics is what you do when there's something to do... strategy is what you do when there's nothing to do. Sounds like you need to work on strategy.

Start with a bit of study on Pawn structure and its influence on middle-game plans.

There's always SOMEthing to do or shoot for.  I'm learning this too.  Look for holes or weak squares you can get a N or B to, preferably on the 6th or 7th rank. . . or block/attack the weak square THEY may be trying to get to. 

  There's always something there... we're just not good enough to see it yet.  Meanwhile read Nim's "My System"  It'll show you many things.  I'm reading it now for the 2nd time, as I wasn't good enough to absorb many things the first time. 

When you cant find anything, or there isnt anything active to do.  Find your least active piece, and improve it.

If you have no pawn breaks and you can't improve your pieces, then you almost certainly messed up earlier in the game.

Chances are though that you can improve your pieces, and you don't realize it.

Here's a position I just made up, but I hope it helps to illustrate:

White to move

White's pieces are developed and centralized. It's time to put the rooks on the open d file which will at least have the benefit of challenging black's rooks... is how some weaker players will probably evaluate this, and rush to play, for example, either rook to d1.

But in fact either rook to d1 gives away white's advantage, and the rook on a1 is better on a1, and the rook on f1 is better on f1.

The a1 rook is well placed because it's ready to support the pawn break a4 and after white captures on b5 (depending on how black recaptures) white will either have infiltration on the a file, or be attacking the weak a6 pawn.
(after a4 black cant capture a4 because too much pressure on the isolated queenside pawns.)

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White has a different plan available here too, and it involves another somewhat sneaky piece in this position... the f3 knight wants to go to h4 where it's looking at the juicy f5 square, and now, white could play the pawn break f4. This is also why the f1 rook is good on f1.

Knight heading to f5 is a common idea when the opponent has moved their h pawn because to remove the f5 knight with g6 means moving another pawn in front of the king, which further weakens the king's position.

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Notice also that black's rooks, doubled on the open file, are totally misplaced. White's bishops cover all the entry points on the d file.

So all this to say, sometimes when you think a piece is well placed it's not and vice versa. Here, the rooks on the open file are bad, the rooks that haven't moved yet are both good, and the centralized knight wants to go to to the edge of the board, Nh4, at least temporarily.

That's misleading... sometimes there is nothing to do, and you're forced to shuffle around and defend for many dozens of moves.