QPR have signed hugely experienced defender Paul Konchesky from Leicester on a season-long loan deal.
Wolves have signed promising young Liverpool winger Sheyi Ojo on a season-long loan deal.
The 31-year-old defender said his farewells in a series of messages posted on his Twitter account.
Bramble tweeted: "I've had a great three years and would like to thank every #SAFC fan for the support they have shown.
"The NE is a special place to play and will always be close to my heart. I wish the club and my teammates all the best for the future."
The news comes as no surprise with the former Ipswich, Newcastle and Wigan man having played a peripheral part for much of the last campaign after an injury-hit spell on Wearside.
Bramble joined the club for an undisclosed fee in July 2010 as then manager Steve Bruce returned to the DW Stadium to snap up a player whose career he had helped to revive during their time together at Wigan.
However, he has made just 51 appearances for the Black Cats during a stay which included a lengthy absence as he fought successfully to clear his name after being accused of two counts of sexual assault.
New manager Paolo Di Canio is currently preparing to reshape his squad after managing to drag the club clear of relegation during a tense conclusion to the campaign, and has given little indication in recent weeks that Bramble had a future on Wearside.