Oztumer, who came through the youth system at Charlton before a spell in the Turkish Premier League, has put pen to paper on a two-year contract after the two clubs agreed a deal.
The 23-year-old scored 33 goals in 49 appearances last season for Ryman Isthmian League outfit Dulwich Hamlet.
"He is a lad that we have looked at, a lot of clubs have looked at him," Peterborough boss Darren Ferguson told the club's official website.
"He is an exciting player, he has great balance, great pace and a typical number 10. I have had a good chat with him and I'm looking forward to working with him."