I haven't ridden the Natchez Trace yet, but I spent several days in New Orleans riding my CB1100. That city's a great place to spend time riding a CB1100 or any bike for that matter.
For starters, the FRENCH QUARTERS is extremely accessible b/c you can always find a FREE parking in between two cars along anywhere. Can't do that w/ a car.
& Uptown St. Charles Ave and the Audubon Park...never a problem w/ parking w/ a bike.
And there's at least a full day of sightseeing and riding just along the RIVER ROAD that enbanks the Miss River in New Orleans; spend one day riding the West Bank Side, the next day riding the east bank side...these are some pics I took along River Road...