A non-reciprocal acoustic device that accomplishes non-reciprocity via linear or angular-momentum bias. The non-reciprocal acoustic device includes an azimuthally symmetric or planar acoustical cavity (e.g., ring cavity), where the cavity is biased by imposing a circular or linear motion of a gas, a fluid or a solid medium filling the cavity. Acoustic waveguides are connected to the cavity or the cavity is excited from the surrounding medium. A port of this device is excited with an acoustic wave. When the cavity is biased appropriately, the acoustic wave is transmitted to one of the other acoustic waveguides while no transmission of the acoustic wave occurs at the other acoustic waveguides. As a result, linear non-reciprocity is now realized in acoustics without distorting the input signal or requiring high input power or bulky devices.