The three-dimensional problem of the modelling of elastic wave propagation in a multi-layered acoustic metamaterial, which is a periodic elastic composite with periodic arrays of interface cracks or planar voids of an arbitrary shape is considered. The boundary integral equation method is extended for this purpose. The unknown crack opening displacement vectors for each array are related using the Floquet theorem and solved using the Galerkin method at the reference delaminations in the arrays. The method provides an efficient tool for fast parametric analysis of the influence of the characteristics of the periodic arrays on the transmission and diffraction of elastic waves. Two modifications of the method are proposed and compared for rectangular cracks. To reduce computational costs, a preliminary analytical evaluation of the arising integral representations in terms of the Fourier transform of Green's matrices and crack opening displacements is presented.