A Josephson diode is a nonreciprocal circuit element that supports a larger dissipationless supercurrent in one direction than in the other. In this Letter, we propose a class of Josephson diodes based on supercurrent interferometers composed of Andreev bound state Josephson junctions or interacting quantum dot Josephson junctions, which are not diodes themselves but possess nonsinusoidal current-phase relations. We show that such Josephson diodes have several important advantages, like being electrically tunable and requiring only time-reversal breaking by a magnetic flux. We also show that our diodes have a characteristic ac response, revealed by the Shapiro steps. Even the simplest realization of our Josephson diode paradigm that relies on only two junctions can achieve efficiencies of up to similar to 40% and, interestingly, far greater efficiencies are achievable by concatenating interferometer loops. We hope that our Letter will stimulate the search for highly tunable Josephson diode effects in Josephson devices based semiconductor -superconductor hybrids, 2d materials, and topological insulators, where nonsinusoidal current-phase relations were recently observed.