Mastoras works with more than 400 Greece-based suppliers, and he said about a dozen of those suppliers have requested he wire money to bank accounts established outside Greece.
"At least a dozen of my suppliers, they gave me accounts in Switzerland and basically Germany, England, where I can wire the money over there because the Greek banking system is almost shut down," he said.
Some products—including Greek olive oil, cheeses, nuts and dried fruits—are sitting idle in Greece storage units. And the problems aren't isolated to idle products. Even if Greek companies want to stomach the risk and ship goods overseas, many can't afford the taxes and expenses to get goods through customs.
If you can scrounge up the money to conduct business, there's a shortage of shipping and packaging materials. Even then, there's not enough fuel, and the transportation system including railways has ground to a halt. The ripple effect of Greece's financial crisis only seems to be getting wider.
Read MoreWhy Greek banks are running out of money
Meanwhile here on U.S. soil, Mastoras and other business owners are watching their inventories dwindle. They're calculating how much supply they have left—maybe a few weeks—before American consumers start to see shortages of their favorite Greek Kalamata olives and filo dough.
"If our stock starts going down, and things start to be scarce, then obviously prices will rise," Mastoras said.
Despite the uncertainty in his home country, Mastoras last week made time to attend the Fancy Food Show at the Javits Center in New York City. Healthy Mediterranean food has grown in popularity among Americans, and Greek food manufacturers make a strong showing at the annual specialty food trade show. Several dozen Greece-based food companies and manufacturers made the trip to New York last week.
Mastoras' customers include food chains such as Whole Foods. Most had one question for him. "Are you going to be able to supply to us?"