Oscillators can be made to run off of any supply. BUt usually they are made to run off of a DC supply.
But oscillators make an oscillating *SIGNAL*. Signals are very low power and you can't drive things with such a low power signal. You would need some kind of driver or amplifier that accepts the oscillating signal and produces a high power output to drive the piezo. It's the same as how you can't drive a motor directly from a microcontroller pin- you use the pin's signal to control a motor driver that produces a high power current to drive the motor. That was probably your problem.