I opened up a microwave and found a buzzer inside. I opened up a buzzer and found this circuit inside. It's very simple as you will see. Only one transistor, 3 resistors and the piezo.
As power is first applied, the piezo is drawing a lot of current through R1 & 2 and shunts it away from the base of the transistor. At some point, as the piezo begins to saturate, the shunting effect diminishes to the point where the transistor starts turning on. This starts discharging the piezo, which is basically a capacitor with a special dielectric. The rate of charge and discharge is also coupled to the deflection of the piezo material, which is also coupled to the air around it and any cavaties which happen to be nearby. This sets up a see-sawing of the mechanical and electrical components at the resonance frequency of this whole "system".
This is about as simple as a piezo driver can get. Because of the feedback coming from the F terminal on the piezo, the circuit will run at the perfect resonance frequency of the piezo. This is the most efficient frequency for a piezo and makes for a very efficient circuit.
I believe the loudness is controlled largely by R3 but you'll have to do your own experimentation ^_-. The way it is drawn it's not very loud, just loud enough to get your attention. After all, you don't really want your microwave as loud as a smoke detector. Or do you?
This is a good circuit for demonstrating the Helmholtz effect without blowing out your eardrums. Most of the piezo elements you will find are enclosed in a Helmholtz resonator already, but you can have fun making your own. This usually takes the form of a chamber sealed to the top of the piezo element. The size of the chamber and the size of the hole are largest contributors to the resonance frequency. The bottom of the piezo, where the leads or contacts are, is often left in some sort of "infinite baffle" arrangement, or is just another vented enclosure. This end has much less to do with the resonance frequency.
One thing to try would be to have a sliding mechanism on the resonator chamber to make it larger and smaller. I bet you can guess what changing the size of the chamber does!
The transistor can be almost anything. Some cross references to the original one are:C944, C945, C1740, C536, C3202, C3198, 2N3901, 2N3974, 2N3975, 2N3976, 2N4424