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.
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
The Chiefs name was derived from a poll held in the Kansas City area.
A.J. Jenkins is number 15 on the Kansas City Chiefs.
Alex Smith is number 11 on the Kansas City Chiefs.
Allen Bailey is number 97 on the Kansas City Chiefs.
Anthony Fasano is number 80 on the Kansas City Chiefs.
Anthony Sherman is number 42 on the Kansas City Chiefs.
Chase Daniel is number 10 on the Kansas City Chiefs.
Cyrus Gray is number 32 on the Kansas City Chiefs.