It is a quite tricky system for Black to handle, especially if he has his heart set on a Mar del Plata style kingside pawn roller. The point of the g4/e4 pawn duo is to meet ...f5 by capturing twice and then trying to plant a piece (preferably a knight) on e4. Then White has ideas of playing f2-f4 himself and/or bringing pressure to bear down the g-file, and suddenly it is Black's king which is under attack, not White's.
I have played this system with White myself and one of the main reasons is that it annoys the hell out of the one-dimensional KID players out there who think that a kingside attack is the be-all and end-all. You have to learn to play in the centre and on the queenside as well, guys!
Incidentally, it is not just in the h3 system that the idea of g2-g4 crops up - it is common also in the Averbakh and Petrosian systems to name but two - indeed, I would argue that thematically these three systems should be grouped together, as White willingly blocks the centre and tries to restrain Black's kingside counterplay.
Even in the Mar del Plata, White is not obliged to sit still and let the pawns come rolling towards him: he can try to throw g2-g4 into the equation there as well to put a spanner in Black's works.
There is no simple solution to Black's problems, but it is nothing to be afraid of, as long as Black knows what he is doing. This is a matter of learning not just specific lines, but typical plans and counter-measures.
Check out the e-books, they will give you some pointers - and go through some games, too, to see what strong Black players do when faced with this sort of plan.
Of course you may have already done this; if so, are there any particular problems you have found?