This just proves there is a problem since serialEventRun() is not defined in the arduino-1.5.1r2 release.
A temporary fix, while waiting for the Arduino company, would be to add a function similar to this
void serialEventRun(void) {
if (Serial.available()) serialEvent();
}
to a sketch that has a serialEvent() function. A true fix needs a lot more added to the IDE for Due.
People are reporting other strange behavior for Serial so you may be out of luck until the bugs are ironed out.