DUBLIN—When Patrick Murphy moved to a small town in western Ireland a decade ago, there were already two other Patrick Murphys in the neighborhood.

They weren't related and lived in different homes. But they all shared the same address: Dromtrasna Collins, Abbeyfeale, County Limerick.

Like many parts of rural Ireland, the town doesn't use house numbers. Some addresses don't even carry street names. And unlike the rest of Europe and most of the industrialized world, Ireland also doesn't have postal codes—the equivalent of a ZIP Code in the U.S.

The Abbeyfeale postman first delivered mail to the Patrick Murphy who had lived in the village the longest, and they worked it out from there.

"My neighbors would get it first," said Mr. Murphy, 40 years old. "They'd have a good read, and they'd go, 'No it's probably not us.' "