Funny question ... not because I've never thought about it, but because it's been a bone of contention with me quite out of proportion to its importance!
Obviously the name of the band changed from
The Grateful Dead to
Grateful Dead very early in their career, so the correct term is not to have the definite article at the start. Unfortunately, "Grateful Dead" just sounds wrong to me.
Jefferson Airplane is equally good to
The Jefferson Airplane, and
Pink Floyd is better than
The Pink Floyd, but
Grateful Dead is just not a good name.
It's got something to do with standard band name morphology: groups with adjective/noun names (such as The Flaming Lips, The Stone Roses, The White Stripes, The Magnetic Fields, The Chemical Brothers, The Golden Palaminos) usually use the definite article; this is true to such an extent that I always think of "The Red Hot Chili Peppers" even though that is not their name.
In practice, I have my iTunes library standardised to
Grateful Dead, but I usually refer to them as
The Grateful Dead on Head-Fi and in conversation. I'm quite ashamed to have thought so much about it, really.