The French Quater, New Orleans
Because parts of it are mostly residential and traffic is rather minimal early in the mornings, the French Quarter provides a terrific setting for a morning run. Moreover, the view of the historic homes, coupled with sights as the cathedral and stores which have yet to open for day, a run in the morning can give you ideas of places to visit during for the remainder of your day.
Starting at Canal you can run up Bourbon Street until you reach Esplanade (which is about 3/4 of mile), turn right and eventually run up Royal and/or Chartres. You can certainly create your own route but I think Royal and Chartres are quite scenic.
Equipment: A good pair of running shoes