Well, in the ASONG AM2302 Product Manual (Page 4, table 4: AM2302 DC Charactersitics) I read that the pull-up resistor has a min and max value (30K-60K) at 5V Vdd.
So I have changed out the 10K I started with for a 47K unit.
But I think I need to hook up the sensor to an ESP-07 module and run a sketch with the DHT library since I have also seen that the timing of the start pulse is really important, both the length of the pulse and the interval between pulses.
Is the data sheet wrong regarding the pull-up R value?