Burnett, whose brother Kaelin is also on Oakland's roster, spent the past two seasons with Miami and has started 59 games over eight years. He had a career-high 110 tackles and 2.5 sacks in 2012.
Burnett said in a conference call with reporters Sunday night that he told the Raiders he wanted to play at a different linebacker spot than his brother, who also plays the position.
"I think the biggest thing was I told him was I wasn't coming in to play the same position as my brother he told reporters, according to the team's website. "I'll go on record saying that. I want to make sure that we all understand that I'm coming in here to assist my brother, first and foremost, and come in here and play the best I can, provide leadership, and ownership of your responsibilities."
Burnett said that teams contacted him soon after he was cut by the Dolphins, who signed linebackers Dannell Ellerbe and Philip Wheeler in the opening days of free agency, leading to the release of Karlos Dansby and himself.
"I heard from teams rather quickly. These were probably the longest four days of my life. It's a very uncomfortable feeling," he told reporters, according to the team's website. "The Raiders, they were in it from day one. This is all a blur for me right now, so I really can't answer that question. It all happened fairly quick, I'll put it that way."
Burnett, a second-round draft pick by Dallas in 2005, spent four seasons with Dallas and two in San Diego before signing with the Dolphins.
Information from The Associated Press was used in this report.