“I need great product designers, and IT people aren’t always great at aesthetics,” says the CIO of one of Europe’s largest retailers at a conference in Berlin recently, describing his hiring challenges. “And I need people who are empathetic and collaborative. I can’t have a great IT architect who has to be locked in a room.” Excuse me? Isn’t that where code writers are most at home: alone in a dimly lit room, a crumpled bag of chips at their side?

“We’re hiring artists, special-effects creators, and people who understand beauty,” says Charles Phillips, CEO of Infor, a maker of enterprise software. We’re at his headquarters in Manhattan’s Silicon Alley, where he’s describing his strategy for competing against industry giants Oracle and SAP. Infor, he says, offers “beautiful business software for your business processes.” This, for software that has long occupied the boiler room of corporate infotech.

The clear trend here is not some fad in the software industry. A mushrooming demand for employees with affective, non-logical abilities spans the economy. Empathy—sensing at a deep level the feelings and thoughts of others—is the foundation. “Non-cognitive skills and attributes such as team working, emotional maturity, empathy, and other interpersonal skills are as important as proficiency in English and mathematics,” reports an advisory group of executives and educators on education reform in the U.K. When author George Anders searched for online job postings that paid over $100,000 a year and specified empathy or empathic traits, he quickly found 1,000 of them from companies as varied as Barclays Capital, McKinsey, and Mars.

It’s happening for several reasons. Partly it’s a search for differentiation in a world where many products and services are becoming commoditized. From computers to refrigerators to websites, they mostly work fine and are reliable enough. How will yours distinguish itself? Intuit’s Smith knows the question to ask: “Did it leave me with a positive emotion?”

More and more, CEOs are concluding that to compete and win, they first need to understand the customer’s inner experience. Their employees need empathy. And that trait is becoming ever more valuable, in part because the supply of candidates who possess it seems to be shrinking—at least in the U.S. Empathy among American college students has declined significantly over the past 30 years, say researchers from the University of Michigan and University of Rochester Medical Center (see chart above). Other research gives little reason to believe it will increase as they grow older.

And that brings us to a deeper issue. We have evolved exquisitely to connect in person. Consider what happens when you’re near someone and his or her face displays an emotion fleetingly, through a so-called micro-expression. Your own face mimics that expression within milliseconds, and the other person, in turn, detects your response. You have empathized without either one of you being aware of it—but it doesn’t happen if you’re alone in your cube. Which, virtually speaking or not, is where many of us spend our time these days. As work becomes dominated by technology, our individual worlds become increasingly cognitive and virtual, lacking in face-to-face contact.

That may be why IT companies are at the vanguard of this new movement in corporate management: Getting ahead in tech today requires, among other skills, seeing the world from outside the cube. Empathy, emotion, and beauty aren’t as easy to measure as other metrics of employee performance. But to use another affect-laden term, it’s time to embrace them.

This story is from the September 22, 2014 issue of Fortune.