Periodic actuation of multiple soft, pneumatic actuators requires coordinated function of multiple, separate components. This work demonstrates a soft, pneumatic ring oscillator that induces temporally coordinated periodic motion in soft actuators using a single, constant-pressure source, without hard valves or electronic controls. The fundamental unit of this ring oscillator is a soft, pneumatic inverter (an inverting Schmitt trigger) that switches between its two states ("on" and "off") using two instabilities in elastomeric structures: buckling of internal tubing and snap-through of a hemispherical membrane. An odd number of these inverters connected in a loop produces the same number of periodically oscillating outputs, resulting from a third, system-level instability; the frequency of oscillation depends on three system parameters that can be adjusted. These oscillatory output pressures enable several applications, including undulating and rolling motions in soft robots, size-based particle separation, pneumatic mechanotherapy, and metering of fluids. The soft ring oscillator eliminates the need for hard valves and electronic controls in these applications.