Because of the strategy involved (honest), curling is sometimes called "chess on ice" (honest).
Visually, though, it reminds me more of another sometime-enthusiasm of mine, the game of go. Both involve the thoughtful placement of stones according to very general patterns.
In go, such patterns are called joseki. They're sort of locally optimal ways for both players to prepare a corner of the go board for the battle for territory that will come later in the game.
In curling, the battle for placement comes a lot sooner, and the choice of viable patterns is restricted by many more factors, a major one being that you can't just lean over a block of cherrywood and pop the curling rock on the exact spot you want it. On the other hand, you are allowed to take out your opponents' stone, if you can.