Until recently most converts came through marriage to an Orthodox man or woman, as in the film My Big Fat Greek Wedding. This film was embarassingly accurate in many details, uncovering some of the less laudable realities of Greek Orthodoxy in America.
The last twenty years or so has seen Orthodoxy capitalizing on the frustrations of Roman Catholics (against contemporary worship) and Protestants (against modern worship and deconstruction of scriptural faith). In Northern Ireland, Orthodoxy capitalizes on the frustrations of both groups with the endless cycle of animosity between them. Protestant clergy visit the Antiochian priest secretly for help that they cannot find elsewhere.
In general, the Orthodox Church doesn't evangelize as an institution. It simply maintains order in doctrine and practice, and disseminates information about itself as a faith community, focusing in on what makes it distinctive. This makes it appear to buy into the ecumenical world view that ‘all paths lead to God,’ which is of course the fundamental heresy of our times.
Real evangelism takes place almost exclusively through the unsupported efforts of ordinary Christians. The style varies, but I would say that what I used to do, read the Bible aloud in public without commenting or preaching, is compatible with the Orthodox tradition.
Orthodoxy is the faith of the martyroi, the witnesses, and that's what we do, wherever we are. No frills, no strings attached, leaving God to do the work of converting men's hearts. All we do is follow along behind Jesus. He does it all. What we can do is catch the fish with the Word of God. He cleans ‘em and sorts ‘em out."
‘Come, follow me,’ Jesus says, ‘and I will make you fishers of men.’ (Matthew 4:19 NIV)
"Come, follow Me", not so much by following Me exteriorly but by loving Me, imitating Me, and I will make you fishers of men, that is, teachers of mankind. In fact, it is with the net of the Word of God that men must be drawn.
— St. John Chrysostom