The adafruit "DC Boarduino" includes a voltage regulator and would operate fine from a 9V battery (not considering possible battery life issues.)
The "USB Boarduino" and stickduino both get 5V power directly from the USB port and would need an extra voltage regulator to work from a 9V battery. (note however, that building a 5V regulator for a 9V battery is a reasonably cheap and easy project.)