McGinty said: "I have enjoyed my time at Edinburgh Rugby and have met some fantastic people. The support from the fans has been magnificent. Good luck to Edinburgh Rugby in the future."
Bradley will continue to coach the side until his contract ends this May, and will share the defensive duties with forwards coach Neil Back, who joined the club in July last year.
Bradley said: "It's been an honour to be Edinburgh Rugby head coach and I'd like to thank my fellow coaches, management team, players, staff, fans and sponsors for their support these past two years.
"Building a strong club with a clear identity, which has shared ownership from the players, management, fans, sponsors and the community at large is the key driver of all successful teams and, for me, it is critical that this work continues and enables both the club and the team to reach its full potential.
"The undoubted highlight of my two years was the joy on the faces of all connected with Edinburgh Rugby when the final whistle went when we overcame Toulouse in last year's quarter-final of the Heineken Cup at Murrayfield.
"I am very disappointed by the run of results this season and will put everything into the remainder of the season to help the club build for a brighter future."
Scottish Rugby's Chief Executive, Mark Dodson, said: "Michael and Billy have played a pivotal role in a number of memorable games at Murrayfield. In particular, last year's journey to the Heineken Cup semi-final in Dublin was a highlight.
"This season's results, however, have been below the standards expected by the club and its fans and we believe the time is right to begin the search for a fresh coaching team."
Edinburgh Rugby managing director, Craig Docherty, added: "On behalf of everyone at Edinburgh Rugby, I would like to thank Michael and Billy for their on and off field contribution and input to the club over the past couple of seasons.
"They were a major part of the landmark achievement of reaching the Heineken Cup semi-final last year and I wish them all the very best for the future."
Edinburgh Rugby will begin the recruitment process for new coaching personnel with immediate effect.
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