Hi fellas, I'm new on the board and found this thread while hunting up noisy GR-20 problems and I see that it ain't just me.  In my setup, (telecaster, GR-20, several stomp boxes, back to GR-20, out to Marshall half stack) I am hearing white noise as mentioned here, and also 60Hz hum.  The problem is definitely NOT a ground loop, and it is NOT the 13 pin cable.  The problem is for sure in the GR-20 effects loop circuitry, and in my (old, experienced, guy) opinion is a shielding problem within the bowels of the GR-20.  The biggest problem I think is that the GR-20 takes an AC input from its transformer so that gets that 60Hz signal all the way into the GR-20's housing.  I tried grounding the bottom metal plate of the GR-20 and that had no effect.  Most of my other stuff takes various DC voltages in, and as long as I keep the wall-warts shielded from the audio, all is well.  The white noise is coming from somewhere else, but I'll bet that it's also caused by poor shielding in the analog audio section of the GR-20.  I'm still trying to quiet this thing down, and if I make any progress, I'll post up here.  Since I've seen this complaint elsewhere, I'd say that the OP does not have a broken GR-20, but a poorly designed one.  I'm gonna poke at Roland a little to see if I can get a response, but I'm not expecting much more than the "check for ground loops" boilerplate fix that everyone gets.  If I come up with an answer, I'll post back here.