That was the 1974 season. Michigan's record that season was 10-1 and they were co Big 10 champions. However, this was at a time when the Big 10 only allowed one conference team to go to a bowl game, that being the conference winner that would go to the Rose Bowl. That edict was discontinued in 1975 and Michigan wound up going to the Orange Bowl while Ohio State went to the Rose Bowl.
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Through the 2008 season, that was the 1981 season when the Seminoles went 6-5. They started 6-2 but lost their final three games (Miami, Southern Mississippi, Florida) by a combined score of 120-36 and were not considered for a bowl game.
Prior to the 2008 and 2009 seasons, that would be the 1972 season through the 1974 season. Michigan played in the 1972 Rose Bowl (1971 season) and their next bowl game was the 1976 Orange Bowl (1975 season).