Adafruit DHT22 Library + Arduino Due not compiling?
There are several Arduino libraries for reading DHT22 and related DHT sensors.  The Adafruit DHT library does not compile for the Due since it uses AVR specific functions not available on ARM (Due).

For the Arduino Due use Rob Tillaart's DHTstable library available from his github page where you can download a zip file.  Note that his project also has a "DHTlib" library which does not support the Due.  There exists yet more DHT libraries which Rob has listed on the DHTLib page on arduino.cc.