Periodic structures involving crossed arrays of cylinders appear as special three-dimensional photonic crystals and cross-stacked gratings. Such a structure consists of a number of layers where each layer is periodic in one spatial direction and invariant in another direction. They are relatively simple to fabricate and have found valuable applications. For analyzing scattering properties of such structures, general computational electromagnetics methods can certainly be used, but special methods that take advantage of the geometric features are often much more efficient. In this paper, an efficient method based on operators mapping electromagnetic field components between two spatial directions is developed to analyze structures with crossed arrays of circular cylinders. The method is much simpler than an earlier method based on similar ideas, and it does not require evaluating slowly converging lattice sums.