Seeing the Preservation Hall Jazz Band in their hometown venue is a nice experience which will make you feel like you are back in the jazz age drinking illegally distilled spirits. That said, it is a very popular tourist venue with two seatings nightly for which you generally need to get there 30 minutes in advance.

If you are there during the day wandering around the French Quarter, particularly along Royal St. and Chartres St., and around Jackson Square in front of the St. Louis Cathedral, you will see street performances every day of the week. This is New Orleans we are talking about, so even the street performers are extremely talented musicians.

In the evening head over to Frenchman St. near Washington Square on the northeast side of the quarter where every club will have its own band playing classic New Orleans style jazz.

For a true treat that is too far from the downtown area for most of the tourists to make it, head over to Oak St. near the western end of St. Charles Ave. (west of the garden district), and check out the Maple Leaf Bar. It feels like a complete dive bar, but some of the most well known bands in the Southeast play there when they roll through town. For a real treat, head to the area early enough to eat at the restaurant next door; Jacques-Imos.

More, from Wikitravel, which - uniquely (as far as I know) for its New Orleans page, has a Listen section:

New Orleans is justly famous for the music it produces. In some other places live music may be thought of as occasional luxury; in New Orleans live music is an essential part of the fabric of life. Parades from the grandest Mardi Gras spectaculars to small neighborhood club events have to have bands to get the locals dancing in the streets. Hey, New Orleans is the birthplace of the "jazz funeral".

There are usually several good performers somewhere in town even on a slow night. Understand that most of the good stuff is not along the tourist strip of Bourbon Street. Even in the parts of town less known to visitors, most sections of the city have at least one (and often several) venue offering great live music.

Budget travelers should know there are usually at least a few free live music events every week in various parks and galleries around town.

The best ways to keep informed about who is playing where and when:

• Gambit's Best of New Orleans, the city's local newsweekly with features on arts and entertainment and whatever else is going on. Gambit also produces the de-facto local restaurant guide and listings.
• Offbeat Magazine is a free monthly local music magazine with extensive listings. Can be picked up at most music venues, coffee shops, and other places around town, or ask your hotel concierge for a copy.
• WWOZ 90.7 F.M. is the community radio station dedicated to local music. At the top of each odd numbered hour they play a listing of the live music happening around town for the day. WWOZ is also good for finding out about special events like "second line" parades and "jazz funeral"s.
• WTUL 91.5 FM is the Tulane college radio station, playing mostly progressive music, but also jazz, classical, and numerous other specialties. At the top of each hour they announce concerts and other events going on around town.

Post-trip update. I had the good fortune of stumbling upon a classical music concert at the Saint Louis Cathedral. The organ was electrifying, as was the rest of the concert.

I also went to Frenchmen Street adjacent to the French Quarter. I listened to a terrific  band at The Three Muses. They were playing Django Reinhardt tunes. 

In sum, the musical heritage of New Orleans lives up to its fine reputation.

Preservation Hall

If you are looking for something that is specific to New Orleans and reflects its character, then Preservation Hall is it. This historic music venue has great jazz with a rotating roster of bands nightly. It may attract tourists, but that is because it is popular rather than touristy.