I borrowed an oscilloscope so I could take a look at the output I was getting from the microphone.  I was originally using the MAX4466 from Adafruit.  At lower frequencies I was getting a fairly well defined waveform, but as I increased the frequency it would soon disappear untill all I could see was noise.

Luckily I had also purchased the MAX9814 that has built in automatic gain control.  With this amplifier I was able to see frequencies up to about 18k.

The amplifier is giving me about 300mV peak-to-peak at this frequency - but my transmitter is only 4 inches away.  I'm thinking I'm going to want to use a circuit/amplifier that is specifically designed for 18k-25k Hz range so that I can get a little more distance between them.