The computational efficiency of FFT-based computational micromechanics is deeply rooted in the underlying regular, that is, Cartesian, discretization. The bottleneck for most industrial applications is evaluating the typically rather expensive constitutive law on the regular grid. In the work at hand, we exploit coarsening strategies to evaluate the material law with the intention of speeding up the overall computation time while retaining the level of achieved accuracy. Inspired by wavelet-compression techniques, we form aggregates of voxels where the local strain tensors are close, and compute the stresses on these coarsened elements. If done naively, such a strategy will lead to intrinsic instabilities whose origin is apparent from a mathematical perspective. As a remedy, we introduce a stabilization technique which is inspired by hourglass control well-known for underintegrated finite elements. We introduce octree as well as sandwich coarsening, discuss the handling of internal variables, report on the efficient implementation of the concepts and demonstrate the effectiveness of the developed technology on simple as well as industrial examples.