I lived in NOLA for 6 years (4 years of Tulane plus two). I didn't bike at the time, but did quite a bit of walking.
I think bike commuting in NOLA would be opposite of what you would do in other cities. In other cities, you look for roads that are less crowded, take side streets, etc. In New Orleans, I think I would do the opposite to a certain extent, especially from Uptown to the CBD. The side streets tend to be a little more dangerous both in terms of crime and potholes. I would NEVER roll a red light in New Orleans and would always look both ways after the light turns green.
For those familiar with the area, a good example is that I would rather take St. Charles downtown than Magazine or Freret.
New Orleans is no less safe than any other major city as long as you keep your head and make wise decisions. The problem is that you tend to more often be too inebriated to do either!