Devices that achieve nonreciprocal microwave transmission are ubiquitous in radar and radio-frequency communication systems, and commonly rely on magnetically biased ferrite materials. Such devices are also indispensable in the readout chains of superconducting quantum circuits as they protect sensitive quantum systems from the noise emitted by readout electronics. Since ferrite-based nonreciprocal devices are bulky, lossy, and require large magnetic fields, there has been significant interest in magnetic-field-free on-chip alternatives, such as those recently implemented using Josephson junctions. Here we realise reconfigurable nonreciprocal transmission between two microwave modes using purely optomechanical interactions in a superconducting electromechanical circuit. We analyse the transmission as well as the noise properties of this nonreciprocal circuit. The scheme relies on the interference in two mechanical modes that mediate coupling between microwave cavities. Finally, we show how quantum-limited circulators can be realized with the same principle. The technology can be built on-chip without any external magnetic field, and is hence fully compatible with superconducting quantum circuits. All-optomechanically-mediated nonreciprocity demonstrated here can also be extended to implement directional amplifiers, and it forms the basis towards realising topological states of light and sound.