Back to the OP and bands... I find it ridiculously hard to answer my favourite, so I'll list a few, as a lot depends on my mood at the time. Some cross over decades, but you get the picture...
1960s: The Doors, The Stones, CCR, The Who, Cream, The Grateful Dead, The Velvet Underground, Them...
1970s: Led Zeppelin, Sly & the Family Stone, George Clinton and P-Funk, Black Sabbath, Jethro Tull...
1980s: New Order, Duran Duran, Depeche Mode, The Cure, U2, The Smiths, Tears for Fears, R.E.M. ...
1990s: Nirvana, Phish, Pearl Jam, Neutral Milk Hotel, Red House Painters, the Chili Peppers, Sublime...
2000s: Blues Traveler, The White Stripes, Radiohead...
Man, I didn't even touch on my other favourites, like Public Enemy, Beastie Boys, Charlie Daniels Band, Lynard Skynard, The Band, The Stokes, Black Eyed Peas...
I know I will hit Submit, and then think of something else.
I guess if I HAD to choose one band, I would say Phish. But variety is what makes all music so great.