We perform a thorough theoretical analysis of resonance effects when an arbitrarily polarized plane monochromatic wave is incident onto a double periodically modulated metal film sandwiched between two transparent media. The proposed theory offers a generalization of the approach that had been developed in our recent papers for the simplest instance of one-dimensional structures to two-dimensional ones. A special emphasis is placed on the films with the modulation caused by cylindrical inclusions; hence, the results obtained are applicable to the films used in the experiments. We discuss a spectral composition of modulated films and highlight the principal role of "resonance" and "coupling" modulation harmonics. All the originating multiple resonances, associated with the surface plasmon-polariton leaky modes, are examined in detail. The transformation coefficients for different diffraction orders are investigated in the vicinity of each resonance. We make a comparison between our theory and recent experiments concerning enhanced light transmittance (ELT) and show the ways of increasing the efficiency of this phenomenon. In the appendixes, we demonstrate a close analogy between the ELT effect and peculiarities of a forced motion of two coupled classical oscillators.