Superconducting thin-film electronics are attractive for their low power consumption, fast operating speeds, and ease of interface with cryogenic systems such as single-photon detector arrays, and quantum computing devices. However, the lack of a reliable superconducting two-terminal asymmetric device, analogous to a semiconducting diode, limits the development of power-handling circuits, fundamental for scaling up these technologies. Existing efforts to date have been limited to single-diode proofs of principle and lacked integration of multiple controllable and reproducible devices to form complex circuits. Here, we demonstrate a robust superconducting diode with tunable polarity using the asymmetric Bean-Livingston surface barrier in niobium nitride micro-bridges, achieving a 43% rectification efficiency. We then realize and integrate several such diodes into a bridge rectifier circuit on a single microchip that performs continuous full-wave rectification up to 3 MHz and AC-to-DC conversion in burst mode at 50 MHz with an estimated peak power efficiency of 60%.