Here are the two places I can recommend because I know the organizations pretty well:

Hands On New Orleans: http://www.handsonneworleans.org/AboutUs/i...
I stayed at Hands On in Biloxi, MS when I was first down there doing recovery work. I stayed with the NOLA Hands On a couple times, they had bunk beds with air mattresses as I recall. Hands On is a secular organization and you can do anything from rebuilding to working with children or cooking for the camp. I'm in my 40's but most of the short-term volunteers were in their 20's. There was usually good music playing and a very laid back atmosphere. Hands On is a national organization that's working out of NOLA and ask for a donation of $20 a day to provide food, and pay for gas and overhead. All tools, safety gear and building materials are provided by corporate sponsors (just FYI if you have a problem with Home Depot or something).

Moving Forward Gulf Coast: http://movingforwardgc.org/17501.html
This is a local grass-roots organization out of Slidell (20 minutes outside NOLA, but where the eye of the storm passed over). You'd be staying with a family, probably Creole, who'd also provide your food. It's unclear from their website if they require any money to work with them. The Executive Director is super-cool and wicked smart, so I know she'll put you to good use (though maybe not rebuilding) and hook you up with nice folks to stay with. All materials are donated by community members and relief organizations.

The only other secular places I can think of are:

Habitat for Humanity: http://www.habitat-nola.org/volunteer/inde...
I only know what they say on their website, but it looks like you can get lodging for $150/week ($30/day) at an elementary school turned into a volunteer camp. You need to buy all your own food.

Common Ground: http://www.commongroundrelief.org /
The most grass roots group, but it looks like (from their web site) they may be filled up the week you're looking to go down, and they might be only looking for skilled labor. But check out the site anyway, I'm only skimming it to give you some basic info.

ACORN: http://www.acorn.org/index.php?id=9943
Their volunteer page doesn't say anything about lodging or food, but you could always ask them.

Good luck finding housing and work, and thank your pointy little head for wanting to go down and help.