The Ducks sent a fourth-round choice and a seventh-round pick in 2015 to the Lightning.
Thompson has scored 85 points in 391 career NHL games with Tampa Bay, Boston and the New York Islanders, winning nearly 52 percent of his faceoffs.
He was an alternate captain with the Lightning season, playing in 81 games while scoring nine goals and excelling on the penalty kill.
The Ducks could have three new centers behind captain Ryan Getzlaf next season after acquiring Ryan Kesler from Vancouver on Friday for Nick Bonino. Anaheim has announced it won't re-sign Saku Koivu, and Mathieu Perreault is a restricted free agent.