Orgeron was fired after the 2007 season when MSU went 8-5 with a Liberty Bowl win. That's the only recent coaching change we've had that you might argue was related to MSU success.
Cutcliffe was fired in 2004 following a 4-7 season and an Egg Bowl win over 3-8 MSU.
Tuberville left for Auburn on his own after the 1998 season.
Brewer was fired after the 1993 season for NCAA violations. Joe Lee Dunn was let go after his one season in 1994 because he was only hired as an interim coach.
Before that, Steve Sloan was fired following the 1982 season when MSU went 5-6. MSU was 9-3 and 8-4 in the previous two years, so you might be able to argue that Sloan's firing was related to MSU, but that would be the only one for us in the last 30 years outside of Orgeron.
And for the record, I think that the changes at MSU were somewhat coincidence. Croom's firing may have been somewhat related to the performance in the 45-0 Egg Bowl last year. Other than that, I think Jackie was let go simply because it was time, not because of our success.
Edited to add: I do think that happens sometimes, and I really think Tuberville's firing in Auburn had as much to do with Bama's 12-0 regular season as it did Auburn's 5-7 season. That, and the general Saban-envy that the Auburn fans have.