The United States captain joined Leeds from RB Leipzig last year and played 24 Premier League games, picking up a hamstring injury in March which ended his season as the club finished 19th and were relegated to the second-tier Championship.
Adams made more than 100 appearances for Leipzig and has won 36 international caps, leading the USA to the World Cup knockout stages last year.
"He is a player we've admired for a long time," Bournemouth chief executive Neill Blake said of Adams, who was linked with a move to clubs including Chelsea and Liverpool.
"It's been widely documented how much his talents have been recognised in this transfer window by other clubs, so to have him sign for Bournemouth shows the ambition we have as a club."
Adams is Bournemouth's seventh signing of the transfer window under new manager Andoni Iraola.
The south coast club have also brought in Dutch winger Justin Kluivert from AS Roma and made Ivory Coast midfielder Hamed Traore's loan permanent.