Locomotion of soft robots typically relies on control of multiple inflatable actuators by electronic computers and hard valves. Soft pneumatic oscillators can reduce the demand on controllers by generating complex movements required for locomotion from a single, constant input pressure, but either have been constrained to low rates of flow of air or have required complex fabrication processes. Here, we describe a pneumatic oscillator fabricated from flexible, but inextensible, sheets that provides high rates of airflow for practical locomotion by combining three instabilities: out-of-plane buckling of the sheets, kinking of tubing attached to the sheets, and a system-level instability resulting from connection of an odd number of pneumatic inverters made from these sheets in a loop. This device, which we call a “buckling-sheet ring oscillator” (BRO), directly generates movement from its own interaction with its surroundings and consists only of readily available materials assembled in a simple process—specifically, stacking acetate sheets, nylon film, and double-sided tape, and attaching an elastomeric tube. A device incorporating a BRO is capable of both translational and rotational motion over varied terrain (even without a tether) and can climb upward against gravity and downward against the buoyant force encountered under water.