In this contribution, a qualitative and quantitative comparison of two- and three-dimensional finite element simulations for magneto-rheological elastomers is presented. Based on a continuum formulation, a microscopic modeling approach is applied. To understand basic deformation mechanisms, the differences between simplified two-dimensional and realistic three-dimensional simulations are initially shown for the example of chain-like structures with different arrangements of the particles. Subsequently, a numerical homogenization is used to connect microscopic and macroscopic quantities: different two- and three-dimensional microstructures are compared with regard to their effective magneto-mechanical behavior.