For the last couple of weeks, the phone at Tasos Ioannidis’s five-star hotel on the breezy island of Mykonos has been ringing steadily, but not with the types of inquiries he wants to field.

European Bank Note

“People are saying they don’t want to confirm a stay or make deposits,” said Mr. Ioannidis, who owns the luxurious Belvedere Hotel, perched on a cliff over the Aegean Sea. “They are afraid of what could happen to their money if Greece leaves the euro and returns to the drachma.”