This paper presents a novel micromechanical procedure for the linear elastic homogenization of composites with periodic microstructures. The procedure is developed for composites with arbitrary number of phases and geometric shapes of the inhomogeneities, in contrast with most existing homogenization approaches. Also, no restriction is made in relation to the mismatch between the properties of the phases and volume fractions of the inhomogeneities. The proposed procedure is based on the Eshelby equivalent inclusion approach and extends a model originally derived for evaluating the effective elastic moduli of periodic two-phase composites. The procedure represents the fluctuating elastic fields within each multiphase repeating unit cell (RUC) using Fourier series, resulting in Lippmann-Schwinger integral equations governing the unknown eigenstrain fields of the inclusions. Unlike traditional iterative algorithms used in Fast Fourier Transform (FFT)-based approaches, the procedure solves the integral equations straightforwardly from a scheme of partition of the domain of each inclusion. The efficiency of the proposal procedure is demonstrated through applications to composites with different arrays of coated fibers and constituent materials.