Born in Paola, Kansas, and raised in Osawatomie, southwest of Kansas City, Dickey led Osawatomie High School to a state championship and graduated in 1967. Its football stadium is named after him, and his number "10" was retired by OHS in January 1971.
Dickey was selected in the third round of the 1971 NFL draft, 56th overall, by the Houston Oilers, where he played four seasons, splitting time with fellow 1971 draft pickDan Pastorini. He and Pastorini were in competition for the starting quarterback role in 1972, and Dickey was having a good preseason. However, he suffered the first in a long line of serious injuries throughout his career, a dislocated and broken hip in the fourth preseason game, which cost him the rest of the year. Dickey started ten games with the Oilers, winning two overall, including Houston's lone win in the 1973 season.
In 1976 he was packaged in a trade to the Green Bay Packers involving John Hadl. His Packers career got off to a rocky start. Dickey suffered a broken leg in 1977, causing him to miss two full calendar years. But by 1980 he was back as the starter. Dickey led the Packers to the 16-team playoffs in the strike-shortened 1982 season. The Packers won the opening round after Dickey went 19 of 26 for 286 yards with 4 touchdowns and no interceptions in a 41–16 triumph over the St. Louis Cardinals. Green Bay was eliminated the following week by the Dallas Cowboys in the divisional round after Dickey went 24 of 37 for 274 yards with 1 touchdown and 3 interceptions as the team fell 37–26. Perhaps the high point of Dickey's NFL career came in 1983 when he powered the Packers' offense to a then-team record 429 points. His 4,458 yards that season served as the team record until being surpassed by Aaron Rodgers in 2011, and threw a career-best 32 touchdowns, the most in the NFL in 1983. Dickey was named second-team All-NFC behind Joe Theismann. Dickey's Packers had beaten Theismann's Super Bowl championWashington Redskins in a thrilling Monday Night Football game earlier that season. The game is included in NFL Network's program "Top Ten Quarterback Duels". Dickey retired from professional football prior to the 1986 season, after he was waived late in training camp. He holds Packers records for highest completion percentage in a game with a minimum of 20 attempts. His record for most passing yards in a 1980 game was broken by Matt Flynn on January 1, 2012 with 480 yards. ; and highest average gain in a season. The latter is an NFL single-season record for quarterbacks with 400 or more attempts.