I'm new to the Church myself, but when I asked this question of people at my church (after reading a little bit about this idea on OC.net, actually; I hadn't previously associated it with any form of Orthodox spirituality), it was explained to me as being foreign to OO tradition. We do
have patron saints, but they're for churches (e.g., the church I go to is named for St. Bishoy, so he is our patron saint, and referred to as such in our prayers), or for people (e.g., this biography of HH Pope Kyrillos VI
mentions that he took the name "Mina" after his patron saint, St. Mina, when ordained as a monk). I'm not sure that we have patron saints for causes as the RCC does. Certainly particular saints are known for certain aspects of their lives that we might relate to at a given point in our own, as Salpy writes, but that's not quite the same thing. In fact, when I mentioned to a friend from church that Catholics pray to St. Anthony of Padua when they have lost something, he didn't believe me. After I convinced him that this is true, he said that we don't do such things, and then went on a bit of a diatribe about how the RCC would be great if it weren't so modern. Oops.