Rogers had spent his first two NFL seasons with Minnesota after being a sixth-round pick in 2002.
A defensive end at Georgia Tech, Rogers has two sacks and four forced fumbles in 43 career games, including 16 starts.
Rogers took the roster spot that became available after the Dolphins released wide receiver Cliff Russell on Tuesday. Russell had been signed last week.
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Newly acquired quarterback Cleo Lemon had a hectic first day with the Dolphins as he began the difficult process of trying to learn the offense as quickly as possible in the event his services become necessary at some point this season.
"(I felt like) a chicken with my head cut off, running around trying to get to the right places," Lemon said in a statement. "The guys helped me settle in and get used to the tempo of practice. The tempo is kind of up-tempo, so I'm just trying to get used to that, fit in and play my role the best I can do it."
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Linebacker Junior Seau and defensive end Jason Taylor didn't practice on Wednesday, and Coach Nick Saban took exception to a Miami Herald report quoting a source that Taylor was likely to play despite his foot injury. Saban said a decision probably wouldn't be made until Friday at the earliest.
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Nose tackle Tim McGill, who was waived last week, has been signed to the practice squad. He took the place of defensive end/outside linebacker Bryant McNeal, who was released.