For what it's worth, we use both rings and rolling in our program.
We like to unroll when the ground is firm or frozen, as it allows all the cows a chance to eat at the same time (and we find that the waste really isn't that much, as we make them clean it up between feedings).
But in the spring time, the ground in the pens and pastures turns into a real morass, so we feed in the rings then.
I've never gone out and raked up what is trampled by feeding in either ring or roll out, but I truly feel it's got to be about equal up here (the cows always seem to knock about a one and a half foot circumference and trample about a foot of it into the muck of hay onto the ground from the ring feeders...and would love to invest in the new kind with the solid material around the bottom, but unless we get stellar prices here this fall, it's just not an option).
Anyway, everybody has their own preferences. Just thought I'd share ours.
Take care.