This work investigates the acoustical properties of a multilayer porous material in which periodic inclusions are embedded. The material is assumed to be backed by a rigid wall. Most of the studies performed in this field used the multipole method and are limited to circular shape inclusions. Here, a mode matching approach, more convenient for a layered system, is adopted. The inclusions can be in the form of rigid scatterers of an arbitrary shape, in the form of an air-filled cavity or in the form of a porous medium with contrasting properties. The computational approach is validated on simple geometries against other numerical schemes and with experimental results obtained in an anechoic room on a rigid grating embedded in a porous material made of 2 mm glass beads. The method is used to study the acoustic absorption behavior of this class of materials in the low frequency range and at a range of angles of incidence.