Probably the best road bike routes in New Orleans are either to follow the Ironman 70.3 New Orleans Bike Course or take the river levee.
For the Ironman bike course head east along the lakefront and then out Chef Menteur Highway to Venetian Isles and back. You can see the course from this map - http://ironmanneworleans.com/bike.html
or to ride the Mississippi River Levee out to Rivertown and back.
The Tammany Trace is an asphalt-paved 31 mile long rails-to-trails conversion on the north shore of Lake Pontchartrain in St. Tammany Parish in Louisiana. The trail is open to bikers and runners or walkers.
There are several trailheads for the trail, listed on this page http://www.tammanytrace.org/find.shtml
* Covington
* Abita Springs
* Tammany Trace Headquarters Trailhead in Mandeville
* Mandeville
* Fontainebleau State Park
* Lacombe
* Slidell-Carollo
The trail will eventually connect to downtown Slidell.