I've just begun playing with the Arduino Due after quite a bit of experience with the Uno and Pro Mini, and either there's something very different about the Due or I have a faulty board.
Basically, I've tried using a 12V DC plug-pack with a center positive output and am getting nothing from the Due -- no power LED, and no flashing pin 13 LED (I am, of course, running "flash" just to get started).
The board works fine as long as it draws power from the USB connection, but the moment I disconnect USB it goes dead.
The only further info I can give is that when I have the board connected only to the plug-pack the 5V rail reads 2.6V and the 3.3V rail reads 1.7V. Connect the USB and they immediately jump to 5.15V and 3.5V respectively.
I'm not sure if this indicates a fault with the board, or with my expectation that the Due should function just as the Uno does (namely, connect an external supply and the board boots and runs the loaded sketch).