Murray was selected by the Baltimore Orioles in the third round of the 1973 amateur draft and had several successful seasons in the minor leagues. He debuted at the major league level on April 7, 1977, and played in 160 games for the Orioles in his first season. In 160 games, he hit for .283 while contributing 88 RBIs, 27 home runs, 29 doubles, and 173 hits while also striking out 104 times during the season. This was the only season in which he struck out over 100 times. He won the American League Rookie of the Year award. In his next season, he played in 161 games while raising his totals, hitting for .285 along with having 95 RBIs, 27 home runs, 32 doubles, 174 hits along with being named to his first All-Star Game and finishing 8th in the MVP balloting. In the 1979 season, he hit .295 along with driving in 99 runs, 25 home runs, 30 doubles and 179 hits in 159 games. Though he was not named to the All-Star Game, he finished 11th in the MVP balloting. He participated in his first postseason. In the 1979 American League Championship Series, he hit 5-for-12 for a .417 batting average along with one home run and 5 RBIs as they beat the California Angels in four games. In the 1979 World Series, he hit 4-for-26 for a .154 average with one home run and 2 RBIs but also four strikeouts as the Orioles lost to the Pittsburgh Pirates in seven games.