We present an efficient method for computing wave scattering by 2D-periodic diffraction gratings in 3D space near cutoff frequencies, at which a Rayleigh wave is at grazing incidence to the grating. At these frequencies (a.k.a. Wood-anomaly frequencies), the spatial lattice sum for the quasi-periodic Green function diverges (the Green function doesn’t even exist!). We present a modification of this lattice sum by images, which results in algebraic convergence. Away from cutoff frequencies, one can actually obtain super-algebraic convergence to the unmodified quasi-periodic Green function by smooth truncation—however, realization of this convergence rate degenerates close to cutoff and one needs to invoke images. This is joint work with O. Bruno, C. Turc, and S. Venakides.