Hey Chris,
Oh, I didn't knew that Z-Diodes produce that much noise as well.
I've build such, too but used a BC547. Basically with C left open, B connected to the Amp stage and E being fed with a tiny current. Actually, it's the voltage level that get's the thing going. The key is to bias the Diode section (whether it may be an actual diode or transistor) in reverse direction until it hits the breakdown limit. What produces the noise is actually thd "current jumping over the barrier". Since this happens not on a constant level but in small peaks it'll produce that broadband white noise.
Funny fact: couple the outputs of two noise generators and hook them up to your home PA. The signal you'll get sounds like you're at the beach of the sea, with waves rushing in again and again.

Kind gereetings,
~Zazzle.