For now, going to Greece is still a fine travel decision, experts said.
"Greece has been in turmoil for a couple of years now, and tourists have never gotten harmed," said Frommer. Greeks continue to welcome tourists, she added.
The U.S. Department of State does not have any current travel alerts or advisories on its website.
The main piece of advice? Bring extra cash and store the euros in a hotel safe.
"Visitors to Greece should be aware of possible banking service disruptions and should bring extra euros and more than one means of payment," the U.K. government said on its website. Greek banks have been closed for more than a week to avoid a massive outflow of money that could lead to their collapse.
Gretel Dennis, a property consultant based in London who visited the Greek island of Corfu earlier this week, said that the island felt safe and removed from the economic tension in Athens.
But she added, carrying "euros was a must and the smaller denomination you had, the more likely you were to be served."
Tourists should also book as much of their plans before arriving in Greece in order to lock down their expenses beforehand and take advantage of a historically good U.S. dollar to euro exchange rate, according to Robert Firpo-Cappiello, editor-in-chief of BudgetTravel.com.