AP Photo

In 1971, Ray Tomlinson sat down at a terminal, typed a bunch of characters on a keyboard, and sent a message to another machine several feet away via a fledgling computer network called ARPANET. More than 40 years later, the rest of us depend on the same basic process—and the same dated technology—used to send that first e-mail.

Most of the innovation since then has happened on top of that old technology. There hasn’t been a big shakeup since the release of Gmail in 2004, which brought threaded messages and a gigabyte of free message storage (an eye-popping amount at the time). By now, many of us are encountering so-called e-mail overload on PCs, smartphones, and tablets. And e-mail shows no sign of...