A diet common in coastal areas of Southern Europe, particularly one with lots of olive oil and nuts, cuts the risk of stroke and other major cardiovascular problems by 30% among high-risk people, according to a new study.

There's a large body of research linking a Mediterranean diet—one heavy on fruits, vegetables, fish and beans—to heart health. But this study, published Monday in the New England Journal of Medicine, is significant both for its size—it followed 7,447 people in Spain over almost 5 years—and its scientific...