The right-hander started 64 games for Tampa Bay from 2009-11, but he was shuttled to the bullpen last season when the Rays had an abundance of starters. He flourished as a reliever, going 3-0 with a 2.43 ERA, creating some flexibility for him in Kansas City.
Davis is due to make $2.8 million this season and $4.8 million in 2014, with the Royals holding options on the next three years.
The jewel of the deal for Tampa Bay is undoubtedly Myers, who turns 22 on Monday.
The power-hitting outfielder batted .314 with 37 homers and 109 RBIs in 134 games at Northwest Arkansas and Triple-A Omaha, and eventually could help provide some protection in the batting order for Rays star Evan Longoria. Myers showed what he could do during the All-Star Futures Game hosted by Kansas City, when he had a pair of hits and drove in three runs at Kauffman Stadium.
He'll finally get a chance to prove it at the major league level at Tropicana Field.
Odorizzi was 15-5 with a 3.03 ERA for Northwest Arkansas and Omaha, and made two late-season starts for Kansas City, going 0-1 with a 4.91 ERA in 7 1-3 innings. Montgomery was once considered one of the Royals' top prospects, but his stock slid last season, when he went 5-13 with a 6.07 ERA last season while getting demoted from Omaha to Northwest Arkansas.
Leonard hit .251 with 14 homers and 46 RBIs in 62 games for short-season Burlington.
"We're excited to add these guys, anxious to get to know them beyond the information we have on them," Friedman said. "I think it's very possible that Wil Myers and Jake Odorizzi will help us win games in 2013, and Mike Montgomery as well."
AP Sports Writer Fred Goodall in Tampa, Fla., contributed to this report.