Do you know what's strange? His speaking voice isn't nearly as nasally as I'd always thought it was. Aside from the usual "thankyouverymuchgoodnight" bit at the end of his shows, I'd never heard him speak until I watched No Direction Home (which I highly recommend).
I went through a phase in my teens where I couldn't stand his voice, but then I realized that's part of what makes him so original. Not only is he America's very best songwriter (name a better one, I dare you--Springsteen is great, Lucinda Williams is great, Willie Nelson is great, but they're no Dylan), but he's never going to have anyone duplicate him and actually pull it off.