An electronic device may be provided with a phased antenna array for conveying millimeter wave signals. The array may be mounted to a substrate that includes transmission line layers having a first dielectric permittivity and antenna layers having a second dielectric permittivity that is less than the first dielectric permittivity. A ground plane may be interposed between the antenna layers and the transmission line layers. The array may be mounted to the antenna layers and transceiver circuitry may be mounted to the transmission line layers. Transmission line traces may be formed on the transmission line layers. The relatively high permittivity of the first set of dielectric layers may allow the transmission line traces to be routed relatively close together with minimal electromagnetic interference. The relatively low permittivity of the second set of dielectric layers may allow the array to operate with satisfactory antenna efficiency, gain, and bandwidth.