The methods through which computational fluid dynamics (CFD) simulations may be solved have evolved rapidly in recent years. One such solution method that has gained traction is the discrete Boltzmann method (DBM), which builds upon the work performed in the development and study of the lattice Boltzmann method (LBM). While the LBM may be used for the efficient simulation of complex flows, it fails to accurately handle flows with complex geometries, such as curved boundaries. As such, the further discretization of the LBM into the DBM, and specifically the finite volume discrete Boltzmann method (FVDBM), has allowed for the development of an alternative to the LBM for complex simulation domains. In this work, a previously developed FVDBM solution on a cell-centered mesh is further studied and validated. The definition of a previously used stencil method is properly provided, and two unique stencils are developed and tested. From this study across flux schemes, simulated problems, and mesh resolutions, generalizations are made for the necessary future development of the FVDBM. Utilizing these insights, a novel three-dimensional FVDBM (3DFVDBM) is developed and validated on a cell-centered mesh. The meshing technique and conversion to three dimensions are outlined for this novel solver, and the validation is performed across a variety of physical problems. After validation, the 3DFVBM is then further verified through a mesh convergence study, and the analysis of several interpolation schemes for the three-dimensional boundary treatment is performed. Through this additional validation and testing, the major sources of error in the 3DFVDBM are confirmed and mitigated, such that the error is fully eliminated in simple cases. The refinement and development of the FVDBM in both two dimensions and three dimensions allows for the future accurate applications of these solvers to real-world problems.