I'd suggest The White Stripes and The Hives, they are pretty accessable and first got big in the time frame we're talking about. I'd also include Black Rebel Motorcycle Club, The Mermen, Cracker, and Tool. That's just some stuff that comes to mind that is somewhat mainstream. Cake was pretty popular around that time, and The Eels did some cool stuff. That's pretty much from a rock genre perspective, I'm sure thyere's other folks on here that could point you toward some great music of other genres. Oh yeah......can't forget Macy Gray, her R&B stuff is awesome, particularly her first one, and Joss Stone is awesome , too, a twenty one year old white chick from England (what is it with them these days, singing American-style soul and R & B?) who sounds like Aretha Franklin.
Hell, there's plenty of good music of all genres and from all times, bt finding it can be really tough sometimes.
And I'll weigh in on the Dead thing while I'm here, they were phenomenal on some nights, lackadaisical on others, and downright awful on some. But the whole scene and experience of going to shows, or touring the country with them, was a really wonderful adventure. Recordings of the phenomenal nights can capture some of the magic, and are worth looking into. A show from Cornell University in Ithaca, NY, in early May 1977 is considered by many to be the "best" show The Grateful Derad ever performed.
I'd also throw in Lucinda Williams to the list, she's pretty fucking awesome, too, and I'm sure there's a ton of other stuff. It's basically a matter of what you like and what genres you are talking about.