We present a Monte Carlo study of the magnetic properties of an Ising multilayer ferrimagnet. The system consists of two kinds of non-equivalent planes, one of which is site-diluted. All intralayer couplings are ferromagnetic. The different kinds of planes are stacked alternately and the interlayer couplings are antiferromagnetic. We perform the simulations using the Wolff algorithm and employ multiple histogram reweighting and finite-size scaling methods to analyze the data with special emphasis on the study of compensation phenomena. Compensation and critical temperatures of the system are obtained as functions of the Hamiltonian parameters and we present a detailed discussion about the contribution of each parameter to the presence or absence of the compensation effect. A comparison is presented between our results and those reported in the literature for the same model using the pair approximation. We also compare our results with those obtained through both the pair approximation and Monte Carlo simulations for the bilayer system. *