I introduce Frustrated Dark Matter models, renormalizable models of Dirac fermion dark matter (DM) that communicates with the Standard Model (SM) through a pair of mediators -- one scalar, and one fermion. Charge assignments of the mediating sector preclude direct coupling of the dark matter to the Standard Model at tree level but couple the SM to dark matter at loop level. I examine the many effective operators generated at one-loop order when the mediators are heavy, and find that they are often phenomenologically relevant. I build a realization with color sextet mediators. I discuss both collider constraints and explore the DM annihilation spectrum which are typically complex in these models. In some benchmarks, I find a fair amount of parameter space left open by experiment and compatible with freeze-out.