The 30-year old goalkeeper departed Rangers this summer after deciding not to switch his contract over to Charles Green's newco and has now signed a two-year deal with the Istanbul outfit.
Scotland international McGregor is now looking forward to getting started with the 13-time Turkish Super Lig champions.
"It's a big club, with a big tradition," he told BJK TV.
"Supposedly, they have great fans and a lot of noise in the stadium so I'm very excited.
"Every time I play, I don't like losing, which is a good thing to have.
"I'll just give my all for the team and hopefully I can do my best."