I have no desire to get into a deep "Shroud" discussion (perhaps they have already happened, and I wasn't around), but I'm not so sure how important it is to the Orthodox. Unlike the multitude of other holy objects of veneration, there is no feast day to commemorate it. There is no entry in any of our Synaraxia or other books discussing the history of the preservation and veneration of the Shroud. There are no hymns anywhere in our service books that talk about its veneration.
There are Orthodox who believe that it is genuine, and those who believe that it is not. I have personally only heard one thing that is convincing at all, and that is the argument that the Orthodox tradition of having a tapestry with an icon of Christ (the epitaphios), which is then taken in procession in Holy Week, may be the remnant of the veneration of the Shroud itself.
And yet, given the flowering of hymnology throughout the Byzantine era, one would have expected that if the Shroud was being so venerated, the hymnology surrounding this would have grown to include specific discussions and reflections on having the actual Shroud of Christ. Every other holy object of veneration -- from the True Cross to the Cincture of the Theotokos -- has this richness of tradition. If the veneration of the Shroud was a living reality, it is inexplicable to me that the silence in our Church's hymnology and writings would be so deafening.
Perhaps Kolokotronis can comment from the Greek perspective -- I may be ignorant of a crystal-clear and well-documented tradition in the Orthodox Church.
"Perhaps Kolokotronis can comment from the Greek perspective -- I may be ignorant of a crystal-clear and well-documented tradition in the Orthodox Church."
I am unaware of anything in Orthodoxy which speaks to the Shroud save a recent article by the uncanonical "Ukrainian Orthodox Church" which apparently came up with some troparia for a joint service with the Uniates. The article was very "in your face" to the MP. I seem to remember that there was some interest in the Shroud in Russian Orthodox circles and there is an alleged sermon by an Archdeacon from Constantinople in the 10th century which refers to the Shroud. It should be noted that the Shroud of Turin is alleged to have been the Shroud of Edessa. At one point there were 43 alleged shrouds kicking around the East. Personally, I hope its real.