I find the idea of time-travelling cloth harder to swallow than traditional accounts of the Resurrection.
The Shroud appeared right after the sack of Constantinople. No doubt it was one of the relics stolen by the Crusaders during the Fourth Crusade. Greece at the time was a Crusader state, and the Shroud I believe was owned by a French noble family that lived in Athens. Eventually it was brought back to France. Now I'm not too sure about all of this, so don't take my word for it.
I read a few years ago that the original 'mandilla' (Veronica/handkerchief) which has the face of Jesus was found in a little church in Italy. The one in the Vatican was merely a copy. No doubt it too was stolen by the Crusaders. When the face of Christ on the Mandilla was superimposed on the face of Christ on the Shroud, it was a perfect match. Since then I haven't read anything about it, so again I might be mistaken.