“Many leisure travelers with iPhones are going to Europe, not turning off their data and allowing their apps to continue downloading data,” said Ken Grunski, chief executive of Telestial, a company that provides wireless services for travelers. “U.S. travelers were already getting hammered with expensive voice calls that start at $1 per minute. I know of people that have returned home to $1,000 or more roaming bills.”

Luckily for tech-savvy travelers who want to stay in touch, there ...