In the presence of QCD axion dark matter, atoms acquire time-dependent electric dipole moments. This effect gives rise to an oscillating current in a nuclear spin-polarized dielectric, which can resonantly excite an electromagnetic mode of a microwave cavity. We show that with existing technology such a “polarization haloscope” can explore orders of magnitude of new parameter space for QCD-coupled axions. If any cavity haloscope detects a signal from the axion-photon coupling, an upgraded polarization haloscope has the unique ability to test whether it arises from the QCD axion.