The Nigeria international midfielder has put pen-to-paper on a four-year deal, which will keep him at Ewood Park until June 2016.
The 30-year-old, who played alongside Danny Murphy at Craven Cottage, becomes manager Steve Kean's seventh signing of the summer.
The Rovers boss said: "Dickson is a player I have admired for a long time and he will be a fantastic addition to the squad.
"He is a powerhouse in midfield, who has played at the highest level with Fulham and we feel confident he will complement our other midfield options."
Dickson started his career at Manchester City, before joining Preston North End in 2002, where he played just short of 150 games.
He joined Norwich City in January 2006, after an initial loan spell, before moving to the Premier League with Sunderland in June 2007.
After just one season at the Stadium of Light, he made the move down to Fulham, where he made 127 appearances, including a starting role in the 2010 Europa League final.
Dickson will fly out to Holland in the morning to join his new Rovers team-mates.