Johnson said the trio heard that they were due to be traded Wednesday and had it confirmed Thursday.
Wood is one of two starting pitchers, along with Miami's Mat Latos, that the Dodgers added to their rotation. The 24-year-old is 7-6 with a 3.54 ERA this season for Atlanta and will make his Dodgers debut Tuesday.
"I was hoping this would be the final stop for me when it was all said and done over the past 48 hours," Wood said. "If there was a team I would ever want to play for, it's here.