We show that a recent proposal for simulating planar hyperbolic lattices with circuit quantum electrodynamics can be extended to accommodate also higher dimensional lattices in Euclidean and non-Euclidean spaces if one allows for circuits with more than three polygons at each vertex. The quantum dynamics of these circuits, which can be constructed with present-day technology, are governed by effective tight-binding Hamiltonians corresponding to higher-dimensional Kagomé-like structures (
n-dimensional zeolites), which are well known to exhibit strong frustration and flat bands. We analyze the relevant spectra of these systems and derive an exact expression for the fraction of flat-band states. Our results expand considerably the range of non-Euclidean geometry realizations with circuit quantum electrodynamics.