Is this endgame position a draw? It’s white to move.

[FEN "8/8/8/3k3p/3P3K/2P3R1/8/7r w - - 0 1"]


I was white, and this was a blitz game which ended up as a draw because I didn't know what to do. My friend and I later analyzed this position, and he suggested that if I had played Rh3! here, then I would've won (of course, if black trades rooks, white wins). But I wasn't convinced, as the trade wasn't forced at all, and judging by the black king's position, I thought black could hold with perfect play (i.e. keep his king where he is, and use rook checks/occupation of the f-file to stop the white king from getting too close to the white pawns). We gave up analyzing after a while, and used an engine to check the position, and it gave +3 to white, which would imply (I think) that there is a forced win. But machines still tend to be wrong, and after my friend played out the line given by Fritz 12, no progress was made by white. The black rook simply occupied the f-file, and the white king couldn't get close to the white pawns. 

Is this a draw? If this isn't a draw, how can white win?
Yes, it seems to be drawn. First consider the following variation, but in particular the position after 2.Kxh5 first:

[FEN "8/8/8/3k3p/3P3K/2P3R1/8/7r w - - 0 1"]

1.Rh3 Rg1 2.Kxh5 Rg8 3.Rf3 Kc4 4.Kh4 Kd5? 5.Rf5+!


Here we don't even need to consult an engine because there is a better oracle to be had. This 6-piece position is covered by existing endgame tablebases, and you can confirm with a query that this is drawn. So we know for a fact that going this 1.Rh3 route leads to a draw. (White could try delaying the pawn capture and making a more productive immediate move, but it doesn't help, and just leads to play like that coming after 1.Kg5, discussed at the end below.)

As to the idea behind Black's play from here, it is to keep the pawns at bay with the king, and as you intimated in your original post, to use the rook to cutoff the white king as much as possible. It's difficult to say much more about the exact drawing method here, because the play can be very complicated (one reason to love endgames, by the way). To give an idea, consider the position after 4.Kh4. 10 of Black's possible moves keep the draw (including those that simply move the rook along the g-file), but the perhaps seemingly harmless 4...Kd5?, putting the king back where it was, is suddenly a loss for Black in 41 moves, but only if White plays 5.Rf5+!, as all other fifth moves again draw. One way to get a good feel for the correct and incorrect play (and for the general notion of how impenetrable optimal play in technical endgames can be) is to toy around with moves at the link I gave above.

White's only other first move option is 1.Kg5, and it's not too surprising that this draws if winning the h-pawn immediately already draws. It's true that White avoids getting his king cut off on the h-file, but the cost of leaving Black with a distracting outside passed pawn is too high. The point is that the black king and rook can easily stop the pawns if the white king isn't helping out his rook. But now if the white king does try to get involved, say via ideas like Kf6-e6 and pushing the d-pawn, Black can keep things level because she has her own passed pawn to play with.