That's great information, guys. With further experimentation, everything you said appeared to be true.

This image from Wikipedia shows that there really is a small reverse current flow, well before reaching the knee; so my diodes really are good, like you said.
But my experimentation showed that using a zener diode to generate white sound in an Arduino project is a very bad idea. The reason is all the digital noise from the Arduino! For starters, the amplification required to bring up the zener noise is so great, it's hard to filter out the digital noise enough before it gets amplified along wtih the desired zenor signal. But beyond that, I discovered the smallest variation in voltage across the zener diode junction (at the knee) from any digital noise, results in a cascade of electrons across the junction far exceeding what would match the voltage change. In other words, the zener diode acts as a powerful high-gain amplifier for any remaining digital noise it sees!
So then I thought about adding a digital "white sound" chip as an alternative, and read about such chips. I learned that:
(1) They use a shift register with feedback to about the 17th register bin to generate random values.
(2) This is also how random number generators work.
(3) Arduino has a random number generator already built in.
BINGO! The built in random number generator, is already the perfect random noise genaerator!
So I wrote some Arduino code to program my AVR to make it's own white sound, and it works perfectly. And better yet, it allows me to vary the loudness of my white sound by simply changing the maximum parameter in the random(max) function.
Here's the part of my code that generates the sound signal:
void whiteBreath() //here
{
boolean up;
byte vol = 0;
while (true)
{
for (byte i=0; i<100; i++)
PORTC = random(vol); // load a random number onto the eight arduino pins
if (up && (vol < 255)) // increase the loudness if not max already
vol++;
else if ((!up) && (vol > 0)) // decrease the loudness if not silent already
vol--;
else
up = !up; // reverse the direction between louder and softer
}
}
void setup()
{
//......
DDRC = B11111111; // sets the eight output pins to be outputs.
// You also could have written the above line as:
// for (byte i=16; i<24; i++)
// pinMode(i,OUTPUT);
// to do exactly the same thing.
//......
}
So now we have the eight bits of the random number output to the Arduino pins. Next voltages on these pins must be combined into an analog voltage, ready for an audio amplifier/buffer for a speaker:

These resistors give each of the eight pins their proper proportion in setting the final output voltage in the sound. The resistor arrangement is used in arduino projects with audio out put, such as this one.
And best of all it WORKS!
Of course, if you use the above, you'll need to change pin numbers and port name to fit the AVR chip you are using. Im now using the Atmega1284p chip.
PS-- An Arduino page that helped a lot in showing me how to do the above is Port Manipulation.