I've found that all inputs that I don't apply a band-stop 45-55 Hz filter to will show very noticable line noise. Even on circuits that are noise filtered before I connect them to the Due (e.g., the Olimex EKG-EMG shields supposedly have a filter circuit built in).
It doesn't seem to matter if it's all running off battery or if I eliminate all power lines on the table - the 50 Hz vibe is literally everywhere.