We investigate ways to generalize Smirnov’s proof of Cardy’s formula, giving the limit of crossing probabilities for critical site-percolation in regions of the usual triangular lattice, to other setups. The natural goal is of course to obtain the same result for general periodic triangulations, and for bond-percolation on the square lattice. As it turns out, it seems to be the case that a priori estimates of the Russo-Seymour-Welsch type are almost sufficient to conclude.