This paper presents the theoretical foundations and the practical implementation of the Fission Matrix Decomposition (FMD) method. The FMD method is a hybrid technique for the rapid and accurate solution of the criticality transport problem in highly heterogeneous media. The method relies on a two-stage sequence, conceptually similar to the approach adopted by production codes, such as CASMO/SIMULATE. First, a database of local fission matrices and coupling coefficients is generated through Monte Carlo calculations. The database is then used to reconstruct the full fission matrix, from which multiplication factor and fission source distribution are computed with a deterministic eigensolver. The FMD method is here tested against two stylized problems: (1) the pressurized water reactor unit-cell problem and (2) the resource-renewable boiling water assembly problem. The accuracy and computational efficiency of the FMD method are compared against the continuous-energy Monte Carlo Fission Source Iteration method, the Fission Matrix-Based Monte Carlo approach, and the lattice-diffusion approximation. For the analyzed cases, the FMD was 100 times faster than diffusion, while maintaining transport accuracy with a mean absolute percent error lower than 1% on the fission source distribution and difference in multiplication factor below 7 pcm.