The beauty of using a pawn to attack a pinned piece is that the protection the pinned piece enjoys becomes irrelevant; your opponent can’t afford to trade one of his pieces for a pawn. But what if you have no such pawns available for the purpose? Then you go after the pinned target with pieces and the question becomes one of comparative might: all hangs on whether you can bring more force to bear against the pinned piece than your opponent can match with defensive reinforcements. Thus your first thought upon pinning a piece (if you can’t take it with a pawn) is to count how many times it's protected and attacked—and how many times it can be protected and attacked if both sides mobilize as fast as possible. Often this involves creative ways of getting pieces into position—perhaps by exchanging the pinned piece for another, where the trade has the side effect of making room for another piece you can use to attack the position.
In the simple example on the left, White’s queen already pins Black’s rook on f7. The rook is protected by its king, so QxR won’t do. Can White apply more pressure against the rook? Not with a pawn, but with a piece: he plays Rb7, and now the pinned target is attacked twice and guarded just once. Black has no way to add protection to the rook; his queen can’t get to the seventh rank without being taken by White’s rook. The rook on f7 is overmatched, and is lost next move.
So that's the analysis of the pin. But in truth White can do even better than taking the rook. He can deliver mate. After 1. Rb7, there are various things Black might do, but he can't prevent White from playing Qg8, and can't escape the mating net that results. (If Black starts with 1. ...g5xf4, don't forget that White can use Be3xf4 as a resource later in the sequence.) Of course the pin still is important, because that is what freezes the f7 rook into place in the beginning and so allows White to get his own rook into the action.