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What makes an Executive Dream Team even better? A superconnected, no-nonsense board of directors.

FORTUNE—A major lesson in the business dramas of recent years is that great executives are almost—but not quite—enough. If the board of directors isn’t right, then nothing’s right. Consider famous failures like Bear Stearns and Lehman Brothers, or costly scandals like News Corp.‘s (NSWA) phone-hacking debacle, in which the boards missed major risks they should have caught. So as we wind up our Dream Team series, we focus on the foundation of any top company: a dream board.

A great board combines several competencies, and well-chosen directors often bring more than one of them. Top of the list now:

Global perspective

It’s amazing how slowly many companies have added genuinely global directors. A foreign passport or a history of exotic vacations doesn’t cut it.

Financial expertise

Sarbanes-Oxley requires that every public company’s board include a designated “financial expert,” and the audit committee has become a director’s most demanding assignment.

Deep understanding of infotech

It’s arguably the largest force shaping global business. Boards need directors who grasp how infotech will change every company’s strategies and business models.

Wisdom, experience, and courage

Two enduring problems on boards are directors who don’t speak up and directors who speak up but don’t know what they’re talking about. The right mix of gray hair, perspective, and assertiveness is worth its weight in gold (even at today’s price).

The contact list of all time

Most companies could use help in broadening their relationships with top leaders in global business and government. Directors can provide it.

Where to find all those traits? Start with global stars like India’s HCL Technologies chief Vineet Nayar, a management visionary, or a Chinese self-made billionaire like Alibaba founder Jack Ma, or a managerial maverick like Haier Group CEO Zhang Ruimin.

Call Hank Paulson. The former Goldman Sachs (GS) boss and Treasury secretary brings finance expertise, government knowledge, long experience, and the cellphone numbers of everyone important. If he can’t reach someone, Condoleezza Rice probably can; she also understands business, having served on corporate boards before becoming national security adviser and secretary of state.

Amazon (AMZN) chief Jeff Bezos may see better than anyone how technology reshapes nontech businesses. For wisdom, experience, and courage, turn to FedEx (FDX) founder Fred Smith, Kraft’s (KFT)Irene Rosenfeld, or recently retired CEOs Doug Conant, who saved Campbell Soup (CPB), James Owens ofCaterpillar (CAT), and Neville Isdell, who righted Coca-Cola (KO) and has deep roots in Africa, a booming market of tomorrow.

Assembling a board with all those members—well, that’s probably dreaming. But hey, it’s a dream team. And it would be a dream in another sense: Shareholders could sleep very soundly.

This article is from the October 17, 2011 issue of Fortune.

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