Straily struck out 124 over 152 1/3 innings in 27 starts in 2013 for the A's, but struggled this season in seven starts, posting a 1-2 record and 4.93 ERA. He made 10 starts with Triple-A Sacramento.
"Let him take a deep breath after the trade, go to Triple-A, get locked in, and hopefully make an impact at the big league level," Cubs president of baseball operations Theo Epstein said of Straily.
The Cubs took the same approach with Jake Arrieta last season when they acquired the right-hander from the Orioles in the Scott Feldman deal.
Russell, a first-round pick in 2012 who was the key piece in the trade, will report to Double-A Tennessee, and he could be in the lineup Sunday. Epstein said it was "a no-brainer for us that we had a chance to get Addison Russell."
McKinney, a first-round pick in 2013, will join Class A Advanced Daytona, where he'll join Albert Almora, the Cubs' No. 1 selection in 2012, in the oufield. Epstein said Almora is a pure center fielder, and McKinney can play all three outfield positions. A left-handed hitter, McKinney was batting .241 at Class A Advanced Stockton.
Carrie Muskat is a reporter for MLB.com. She writes a blog, Muskat Ramblings, and you can follow her on Twitter @CarrieMuskat. Daniel Popper is an associate reporter for MLB.com. He also can be found on Twitter @danielrpopper. This story was not subject to the approval of Major League Baseball or its clubs.