Within their division (the NFC South), the Atlanta Falcons are the Saints' biggest rival. They don't have a major ongoing rivalry with any one team outside their division unless you count the Chicago Bears, who beat the Saints in the NFC Championship game and kept them from making the Super Bowl after the 2006 season.
The New Orleans Saints, currently and pre-NFC South days. It's always been the Saints at #1 for rivals.
I believe the Oakland Raiders are the Kansas City Chiefs biggest rival. The Chiefs lead the all time series 59-51-2 as of the 2013 season and lead 2-1 in playoff matches.
Kansas City Chiefs cheerleader, Susie Derouchey
Kansas City, Missouri.
Arrowhead Stadium, home of the Kansas City Chiefs, is located at One Arrowhead Drive, Kansas City, Missouri, Jackson County
Jeff Allen is number 71 on the Kansas City Chiefs.
Alex Smith is number 11 on the Kansas City Chiefs.
Eric Kush is number 64 on the Kansas City Chiefs.
Tyler Bray is number 9 on the Kansas City Chiefs.
The Chiefs name was derived from a poll held in the Kansas City area.
Frank Zombo is number 51 on the Kansas City Chiefs.
Demetrius Harris is number 47 on the Kansas City Chiefs.
Anthony Sherman is number 42 on the Kansas City Chiefs.