Hi,
I tried a sheild a few days ago, It will not work as the SPI pins on the due are not connected through the normal digital pins as they are on UNO and MEGA, they are instead broken out through a six pin male head in the center of the board - see the pinout.
I made a little adaptor to connect my sheild to this, for me this also did not work - I am guessing that it was down to the level shift IC on the breakout board which is designed to shift down from 5V to 3.3V, the due is already at 3.3 volt.
The last thing I tried, was to solder jumpers directly to the SD Card and connect these to the SPI pins of the Due, this worked very well - it also means I can't easily connect my SD Card back to a PC, but that may or may not be a problem for you - it is not a problem in my application - as an alternative, you might be able to by pass the level shift IC on your breakout board.
The pin out of the SD Card is in a link in the
http://elasticsheep.com/2010/01/reading-an-sd-card-with-an-atmega168/
Duane B