If you’re a hard-core tennis lover, it’s hard to take a sport called pickleball seriously. There’s the funny name. You serve underhand and hit something that looks like a Wiffle ball. Ken Curry snubbed the game for years. “I thought it was a geezer sport,” he says.

Dr. Curry, a cardiologist in Kennewick, Wash., has tennis bona fides. He played on the Colorado State University-Pueblo team, and after graduation he postponed medical school to pursue a tennis career that lasted 1½ years. In his prime, he held a world ranking...