Spoke too soon... not a power supply issue.
The issue occurs when the Due is connected via the USB programming port, and is sending serial data, but there is no serial monitor program open to receive the data.
If either the serial monitor in the iDE, or something like Putty outside the IDE, is open and connected to the programming USB virtual serial port, no resets.
Close the serial monitor, and the resets occur. It seems to be an interaction with the USB port (windows 7 x64) and the Due. A reset occurs when the rx buffer is full on the USB port perhaps?