We propose a new strategy to search for dark matter axions in the mass range of 40-400 μeV by introducing dielectric haloscopes, which consist of dielectric disks placed in a magnetic field. The changing dielectric media cause discontinuities in the axion-induced electric field, leading to the generation of propagating electromagnetic waves to satisfy the continuity requirements at the interfaces. Large-area disks with adjustable distances boost the microwave signal (10-100 GHz) to an observable level and allow one to scan over a broad axion mass range. A sensitivity to QCD axion models is conceivable with 80 disks of 1 m^{2} area contained in a 10 T field.