There is a variety of microstructured materials that involve voids and pores, for example, high-porosity foams, mechanical metamaterials, or composites involving defects due to damage and cracking, respectively. Computational methods based on the fast Fourier transform (FFT) typically face convergence problems for such microstructures unless specific discretizations are used, most prominently the discretization on the staggered grid. FFT-based methods were originally developed for periodic boundary conditions, and recent work provided extensions to Dirichlet and Neumann boundary conditions on the unit cube faces by utilizing dedicated sine and cosine series. Unfortunately, such approaches were only developed for discretizations that fail to converge for complex porous microstructures. The article at hand closes this gap by constructing the appropriate Eshelby-Green operator for the displacement gradient associated with the staggered grid discretization and Dirichlet boundary conditions. The eponymous staggering of the displacement variables infers certain challenges to be resolved, that is, the construction is significantly more difficult than for the cases discussed in the literature. However, our innovative techniques permit treating the class of microporous materials—which have a wide range of applicability—in a robust and efficient way. We showcase the superiority of the novel techniques via dedicated computational experiments.