Electrical – When oscillating a piezo, should I use AC or DC square wave

A tilt sensor would probably do. All the large component vendors stock varying types of these.

A simple DIY solution might be something like a small spring (from a pen?) placed around a stiff wire so any disturbance causes the spring to contact the wire and close the circuit (connections at bottom of spring and wire like a SPST switch)

The above would obviously need a small battery/solar cell/energy harvesting solution of some sort. If you are trying to do this without a battery then I don't think it's possible with a piezo, as they only produce tiny amounts of current. A shake to charge torch type circuit might be made to work (e.g. magnet slides through coil on tilt)

EDIT - With the requirement of no battery now confirmed, I think I would try the coil idea mentioned above. If all you need is a brief flash then when the thing is tilted and the magnet slides through the coil it should work, You can downsize as necessary (within reason)
Couple of links:
DIY Shake flashlight
Shake Flashlights - how they work
This will likely be pretty difficult to do well with no "intelligence" or energy reserve (e.g. cap charged very slowly with something like a tiny solar cell, discharged through LED on spring sensor closing)

Try these linear amplifiers made by Apex. They are designed specifically for ultrasound aplications.