Adcock was designated for assignment last Thursday to make room on the 40-man roster for infielder Edinson Rincon.
Selected in the Rule 5 Draft from the Pirates in 2010, Adcock jumped from Class A Bradenton to Kansas City, where he spent the entire 2011 season.
"It was a big jump, but I just treated it just the same: 60 feet, 6 inches," he said. "The hitters are better, but the swings and misses aren't really different. Bigger stadiums, more fans, but I tried to treat it just the same and go out there and do my job."
He was 1-1 with a 4.62 ERA in 24 games (three starts).
"Not too many people in baseball get that opportunity, and I'm really grateful for it," Adcock said. "For what the Royals' organization did was make it possible for me to build a new home for me and my wife, Samantha."
Last year, he shuttled between Triple-A Omaha and Kansas City. In 12 games with the Royals, he was 0-3 but had a 2.34 ERA. This year he was 3-4 with a 7.09 ERA in 10 games for Omaha.
"It started good, and I felt confident," he said. "My last five starts, or whatever it was, have been kind of a struggle.
"I'm really happy and really excited about the new opportunity with the Diamondbacks. I'm going to pitch on Sunday, but I'm not sure if its relief or a start, but we'll see, and whatever it is, I'm going to take it and run with it."