Traditional New Orleans Funerals feature sad music before and during the funeral, and happy dixieland jazz on the way back. The second line were usually people dancing and following the band (behind the first line). Check out some photos of a new orleans funeral.
The drummers would march with snare and bass drums playing New Orleans drum beats like the ones shown below. The grooves are always heavy on the snare with lots of offbeat accents. I think of it as a funky march. The bass drum rhythms are never straight four-to-the-floor. They also emphasize offbeat rhythms. One way that this genre is different from most jazz genres is that the feel isn't really syncopated - the eighths are almost completely straightened out.
Be sure to check out the Dirty Dozen Brass Band for a great example of the groove.