Which band IS really America's Greatest (rock & roll band)?

I would vote for the Grateful Dead as well.   The band is a melting pot of musical influences.  At the time of their formation here was the lineup:

Jerry Garcia-Accomplished bluegrass banjo player.
Bob Weir-Folkie
Pigpen-Blues singer and harp player
Phil Lesh-Trumpet player and classical composer
Bill Kreutzmann-Jazz drummer
Mickey Hart-Studied with Buddy Rich and Gene Krupa so jazz is in his DNA as well.

It all came together to form one of the most unique and long lasting bands in history.  

marqmike, Traffic was a British band. At least Winwood was British. 

whart, on what basis do you say that The Band was not an American band? I generally consider Canada to be American. Maybe "American" just means U.S.A.? That would take The Band out of consideration.

Not sure why anyone would say the Eagles were not rock and roll. They got most of their air time on rock stations and I'm pretty sure they primarily charted on rock/pop charts. They did chart on country charts too but I doubt anything on Hotel California would be considered anything but rock. Not defending the Eagles. They were never my cup of tea but if they weren't rock what were they?

MC5

REM, the stooges, Nirvana, stone temple pilots, Pearl Jam, the replacements, soundgarden, Red Hot Chili Peppers, the doors, queens of the Stone Age, velvet underground and of course......Chicago!

@n80- not to be picayune, but I generally don’t refer to Canada as America, even though we are on the same continent. The Band were great, love ’em.
To me, rock and roll is the stuff that came out of the ’50s as a mix of boogie, country, blues and other forms- represented by that long list of notable performers from the era. I think that it pretty much died by the time the Eagles came around; their first two albums, which were country rock, were good- then, they became a sort of stadium rock act.
None of it was rock and roll in the narrow sense that I’m talking about- Jerry Lee Lewis, Live at Hamburg, for example, is a mind blower--not released here in the States at the time due to his PR problems with marrying his young cousin.
Some of the bands mentioned (though not necessarily American) did dive into rock and roll on occasion, but acts like the Beatles (who did r/r covers early on) and the Stones (no matter how much they ’rock’) weren’t in my estimation, mainly "rock and roll."
If the question is best rock band, different answer.
But, you know like I do there is no ’best.’ Today, I spent time listening to "Barbed Wired Sandwich," Black Cat Bones, a precursor to the band Free-- delicious, biting blues rock. Love it, but it ain’t "rock and roll" to me.
Bill