I have to go with 1984 BYU. They are one of only two teams in history that were awarded the National Championship but did not play in either the Rose, Sugar, Orange, Fiesta, or Cotton Bowl.
Michigan was unranked going into the Bowls that year and had lost at home to Washington 20-11, early in the season. Washington was ranked #1 after weeks 7-10 before losing at USC 16-7. Then Nebraska was #1 before losing at home to Oklahoma 17-7. Oklahoma jumped up to #2 behind BYU after the win.
Final Polls Before Bowl Games
UPI-AP
1......1. BYU.................12-0-0
2......2. Oklahoma...........9-1-1
6......3. Florida...............9-1-1
3......4. Washington.......10-1-0
4......5. Nebraska............9-2-0
5......6. Ohio St..............9-2-0
Holiday -- #1 BYU over Michigan 24-17
Orange -- #4 Washington over #2 Oklahoma 28-17
Sugar -- #5 Nebraska over #11 LSU 28-10
Rose -- #18 USC over #6 Ohio St. 20-17
#3 Florida didn't go to a Bowl due to probation.
If strength of schedule had been factored in to the polls that season, I don't think BYU would've been ranked in the top five. The Orange Bowl should've been for the National Championship.
1984-Brigham Young (13-0-0) (WAC)
9/1 @ Pittsburgh (3-7-1) W 20 14
9/8 vs. Baylor (5-6) W 47 13
9/15 vs. Tulsa (6-5) W 38 15
9/22 @ *Hawaii (7-4) W 18 13
10/6 @ *Colorado State (3-8) W 52 9
10/13 vs. *Wyoming (6-6) W 41 38
10/20 @ *Air Force (8-4) W 30 25
10/25 @ *New Mexico (4-8) W 48 0
11/3 vs. *Texas-El Paso (2-9) W 42 9
11/10 vs. *San Diego St. (4-7-1) W 34 3
11/17 @ *Utah (6-5-1) W 24 14
11/24 vs. Utah State (1-10) W 38 13
12/21 vs. Michigan (6-6) W 24 17 @ San Diego, CA Holiday Bowl
1984-Michigan (6-6-0) (Big Ten)
9/8 vs. Miami (Florida) (8-5) W 22 14
9/15 vs. Washington (11-1) L 11 20
9/22 vs. *Wisconsin (7-4-1) W 20 14
9/29 @ *Indiana (0-11) W 14 6
10/6 vs. *Michigan State (6-6) L 7 19
10/13 vs. *Northwestern (2-9) W 31 0
10/20 @ *Iowa (8-4-1) L 0 26
10/27 vs. *Illinois (7-4) W 26 18
11/3 @ *Purdue (7-5) L 29 31
11/10 vs. *Minnesota (4-7) W 31 7
11/17 @ *Ohio State (9-3) L 6 21
12/21 vs. Brigham Young (13-0) L 17 24 @ San Diego, CA Holiday Bowl
1984-Washington (11-1-0) (Pac 10)
9/8 vs. Northwestern (2-9) W 26 0
9/15 @ Michigan (6-6) W 20 11
9/22 vs. Houston (7-5) W 35 7
9/29 vs. Miami (Ohio) (4-7) W 53 7
10/6 @ *Oregon State (2-9) W 19 7
10/13 @ *Stanford (5-6) W 37 15
10/20 vs. *Oregon (6-5) W 17 10
10/27 vs. *Arizona (7-4) W 28 12
11/3 vs. *California (2-9) W 44 14
11/10 @ *Southern California (9-3) L 7 16
11/17 @ *Washington State (6-5) W 38 29
1/1 vs. Oklahoma (9-2-1) W 28 17 @ Miami, FL Orange Bowl
1984-Oklahoma (9-2-1) (Big 8)
9/8 vs. Stanford (5-6) W 19 7
9/15 @ Pittsburgh (3-7-1) W 42 10
9/22 vs. Baylor (5-6) W 34 15
9/29 vs. *Kansas State (3-7-1) W 24 6
10/13 vs. Texas (7-4-1) T 15 15 @ Dallas, TX
10/20 @ *Iowa State (2-7-2) W 12 10
10/27 @ *Kansas (5-6) L 11 28
11/3 vs. *Missouri (3-7-1) W 49 7
11/10 @ *Colorado (1-10) W 42 17
11/17 @ *Nebraska (10-2) W 17 7
11/24 vs. *Oklahoma State (10-2) W 24 14
1/1 vs. Washington (11-1) L 17 28 @ Miami, FL Orange Bowl
Michigan was unranked going into the Bowls that year and had lost at home to Washington 20-11, early in the season. Washington was ranked #1 after weeks 7-10 before losing at USC 16-7. Then Nebraska was #1 before losing at home to Oklahoma 17-7. Oklahoma jumped up to #2 behind BYU after the win.
Final Polls Before Bowl Games
UPI-AP
1......1. BYU.................12-0-0
2......2. Oklahoma...........9-1-1
6......3. Florida...............9-1-1
3......4. Washington.......10-1-0
4......5. Nebraska............9-2-0
5......6. Ohio St..............9-2-0
Holiday -- #1 BYU over Michigan 24-17
Orange -- #4 Washington over #2 Oklahoma 28-17
Sugar -- #5 Nebraska over #11 LSU 28-10
Rose -- #18 USC over #6 Ohio St. 20-17
#3 Florida didn't go to a Bowl due to probation.
If strength of schedule had been factored in to the polls that season, I don't think BYU would've been ranked in the top five. The Orange Bowl should've been for the National Championship.
1984-Brigham Young (13-0-0) (WAC)
9/1 @ Pittsburgh (3-7-1) W 20 14
9/8 vs. Baylor (5-6) W 47 13
9/15 vs. Tulsa (6-5) W 38 15
9/22 @ *Hawaii (7-4) W 18 13
10/6 @ *Colorado State (3-8) W 52 9
10/13 vs. *Wyoming (6-6) W 41 38
10/20 @ *Air Force (8-4) W 30 25
10/25 @ *New Mexico (4-8) W 48 0
11/3 vs. *Texas-El Paso (2-9) W 42 9
11/10 vs. *San Diego St. (4-7-1) W 34 3
11/17 @ *Utah (6-5-1) W 24 14
11/24 vs. Utah State (1-10) W 38 13
12/21 vs. Michigan (6-6) W 24 17 @ San Diego, CA Holiday Bowl
1984-Michigan (6-6-0) (Big Ten)
9/8 vs. Miami (Florida) (8-5) W 22 14
9/15 vs. Washington (11-1) L 11 20
9/22 vs. *Wisconsin (7-4-1) W 20 14
9/29 @ *Indiana (0-11) W 14 6
10/6 vs. *Michigan State (6-6) L 7 19
10/13 vs. *Northwestern (2-9) W 31 0
10/20 @ *Iowa (8-4-1) L 0 26
10/27 vs. *Illinois (7-4) W 26 18
11/3 @ *Purdue (7-5) L 29 31
11/10 vs. *Minnesota (4-7) W 31 7
11/17 @ *Ohio State (9-3) L 6 21
12/21 vs. Brigham Young (13-0) L 17 24 @ San Diego, CA Holiday Bowl
1984-Washington (11-1-0) (Pac 10)
9/8 vs. Northwestern (2-9) W 26 0
9/15 @ Michigan (6-6) W 20 11
9/22 vs. Houston (7-5) W 35 7
9/29 vs. Miami (Ohio) (4-7) W 53 7
10/6 @ *Oregon State (2-9) W 19 7
10/13 @ *Stanford (5-6) W 37 15
10/20 vs. *Oregon (6-5) W 17 10
10/27 vs. *Arizona (7-4) W 28 12
11/3 vs. *California (2-9) W 44 14
11/10 @ *Southern California (9-3) L 7 16
11/17 @ *Washington State (6-5) W 38 29
1/1 vs. Oklahoma (9-2-1) W 28 17 @ Miami, FL Orange Bowl
1984-Oklahoma (9-2-1) (Big 8)
9/8 vs. Stanford (5-6) W 19 7
9/15 @ Pittsburgh (3-7-1) W 42 10
9/22 vs. Baylor (5-6) W 34 15
9/29 vs. *Kansas State (3-7-1) W 24 6
10/13 vs. Texas (7-4-1) T 15 15 @ Dallas, TX
10/20 @ *Iowa State (2-7-2) W 12 10
10/27 @ *Kansas (5-6) L 11 28
11/3 vs. *Missouri (3-7-1) W 49 7
11/10 @ *Colorado (1-10) W 42 17
11/17 @ *Nebraska (10-2) W 17 7
11/24 vs. *Oklahoma State (10-2) W 24 14
1/1 vs. Washington (11-1) L 17 28 @ Miami, FL Orange Bowl