I just hit the exact same issue with my Arduino Micro. And, I have a solution!
The RF24 library uses digital pins 11, 12 and 13 for SPI communication with the nRF24L01. However, on Arduino Micro, the SPI pins are different! If you check the documentation http://arduino.cc/en/Main/arduinoBoardMicro, you'll find this:
Note that the SPI pins are not connected to any of the digital I/O pins as they are on the Arduino Uno, they are only available on the ICSP connector and on the nearby pins labelled MISO, MOSI and SCK.
Essentially, instead of connecting the nRF24L01's pin 5, 6 and 7 to Arduino's 11, 12 and 13, connect it to the SCK, MOSI and MISO pins:
| Arduino Micro --> | nRF24L01 |
|---|---|
| SCK | 5 |
| MOSI | 6 |
| MISO | 7 |