Registering to Vote without Proof of Residency
May I Please Register to Vote Now?
While you have the right to vote, as a US Citizen, RSA 654-12 section B details how those without "proof of residency" (i.e., if you opted not to have your mailing address printed on your DMV ID) can register.
You'll need a photo ID. They accept virtually anything though, from a DMV ID to something you printed up at Kinkos and had laminated the night before that looks official, i.e., an ID for your own business, they really don't care, so long as it's current. Bring along your birth certificate just to be extra sure. There's a blue form you sign, the woman must recite an oath with your right hand raised which you say "yes" to (where you've sworn that you're telling the truth), and then you're registered.
By the way, your voter registration is not valid for proof of residency. I've tried this, and talked to others who also failed at doing this. Apparently a legal document which is tracked and recorded carefully is less legitimate than a hand written (but notarized) piece of paper which is not tracked.