Tough to not name 20 or more but to give one of my favorites right now that don't have a Stones sound I'd say The Mavericks. "Too rock for country and too country for rock." Not one of today's bro-country bands, they been around for quite awhile now.
beat me to it...was gonna say i'm seeing brian wilson in a few hours in bristol. how about the miles davis band from 1959 ? not a lot of rolling stones going on in that.
Well, they are not bands, but i like also.. john Hiatt,John mellencamp,and as a band de dijk,skotwal also golden earring, steve earle,david sylvian, etta james, jeroen
Don't know about 'band' cos popular music can be so interconnected is some way.. But the most non-stones music I've loved is anything pre-55: Sinatra, Judy Garland.... show tunes! Old Hollywood musicals etc...