An electronic device may be provided with a dielectric cover layer, a dielectric substrate, and a phased antenna array on the dielectric substrate for conveying millimeter wave signals through the dielectric cover layer. The array may include conductive traces mounted against the dielectric layer. The conductive traces may form patch elements or parasitic elements for the phased antenna array. The dielectric layer may have a dielectric constant and a thickness selected to form a quarter wave impedance transformer for the array at a wavelength of operation of the array. The substrate may include fences of conductive vias that laterally surround each of the antennas within the array. When configured in this way, signal attenuation, destructive interference, and surface wave generation associated with the presence of the dielectric layer over the phased antenna array may be minimized.