Oddly, the work we Orthodox do (ably described in kosta's reply) manages to get the sharing Christ work done, too, though without any obvious mechanism:
Siberia and Alaska were largely converted by monks who didn't go out to preach the Gospel, but to find a northern equivalent of the desert so beloved of the first monastics. Many native Alaskans attribute their conversion to St. Herman, a hermit (!)
And, we now have quite active mission fields in Africa (and in rechristianizing Albania and to a lesser extent Russia). In subsaharan Africa, though the spread of Orthodoxy got its start in what, from the time of the conversion of the Rus, has been a peculiarly Orthodox way: people came looking. As St. Vladimir went looking for the true monotheistic faith and concluded it was Holy Orthodoxy on the basis of our liturgy in its most glorious expression, in East Africa Christians whose ancestors had converted by protestant missionaries began wondering which church was the True Church, and on the basis of patristic and historical studies concluded Holy Orthodoxy was it.
Both stories have been repeated both individually and by groups: converts come to the Church by wandering in and being smitten by the beauty of Orthodox worship (Bishop. Dmitri of the OCA, for example, was raised as a Baptist, but started sneaking off to an Orthodox church as a youth--his sister is now an Orthodox nun) or by studying the history of Christianity as a whole and concluding the same thing the East Africans did (Jaroslav Pelikan--Memory Eternal!--the foremost church historian of the 20th century and the body of protestant evangelicals who converted under the leadership of Fr. Peter Gilquist, being examples).
That's interesting, thanks. All I knew about Orthodox evangelism was the latter. :) I am very glad to hear that you have active missionaries. I didn't know that.