An equivalent-network model is here proposed to characterize two-dimensional planar periodic arrays of arbitrary scatterers/apertures embedded in a layered environment. The model is an extension of the approach previously developed by some of the authors, which only considered simple rectangular scatterers. A key underlying assumption in the present approach is that the current/field distribution in the scatterer can be factorized so that the spatial profile is independent of the frequency in the considered range of interest. This approximation is proven to work properly for a great variety of useful planar scatterer/aperture patterns, even at frequencies within the diffraction regime. The spatial current/field profile is determined from a full-wave simulation at a single and low frequency value. Our numerical results are validated through comparison to commercial simulators for very wide frequency ranges as well as with previously proposed circuit-model approaches.