© 2019 The Authors Motivated by the problem of acoustic plane wave scattering from an infinite periodic array of cylindrical scatterers, we present a new and easily-implemented way of calculating the quasi-periodic Green's function. This approach is based on an asymptotic expansion of the summand in the quasi-periodic Green's function in order to derive a tail-end correction term, allowing for a rapid and accurate approximation of the function. The tail-end approximation is shown to have much better and faster convergence properties than the usual truncation approach and competes very well with state-of-the-art alternative techniques. This method is then combined with a boundary element scheme to calculate the transmission and reflection coefficients associated with arrays of cylinders of different cross-sections and varying aspect ratios. The results are validated against the existing literature and by independent finite element calculations.