I was walking home in the pouring rain with my six-year-old. She complained about getting wet, while refusing to use her umbrella. I suggested she run home to get less wet. but she figured she would get more wet.
By the Numbers
Seems that lots of people with bigger brains or budgets than me have taken a run at this question. For an armchair approach, here is a mathematical formulation that suggests the result will depend on the direction of the rain. In the Flying Circus of Physics book, Jearl Walker assumes you're wearing a rain hat so you don't have to worry about your head getting wet. If the rain is toward your front or directly overhead, then running the faster the better. But if the rain is on your back, you should somehow run with the rain, with a speed equal to its horizontal velocity. Jay Ingram in the Science of Everyday Things adds that if you slant your body to minimize the surface getting hit by the rain affects can affect how wet you get as well. This interactive page lets you input different parameters, including your speed and dimensions as well as the angle and speed of the rain to see how wet you will get.
Testing, 1, 2
Cecil Adams of Straight Dope fame claimed the number of drops that land on your head will depend on the amount of time you spend in the rain but the amount that hits your chest is the same whether you walk or run. And he supposedly tested this out. Similarly sized, British climatological jogging buddies, Thomas Peterson and Trevor Wallis, compared the effects of running and walking by weighing identical clothing before and after one ran and the other walked 100 metres in the rain. The clothes of the runner (Wallis) was forty percent drier. TV's Mythbusters suggested it would be better to walk in their first look at it. But later they redid it, and concluded it was better to run.
So I had more or less taken the scientific path to this siuation, but I realize my daughter, who studies karate, may have been following the intuition of a martial artist. According to Tsunetomo Yamamoto in the Hagakure: the Book of the Samurai,
"There is something to be learned from a rainstorm. When meeting with a sudden shower, you try not to get wet and run quickly along the road. But doing such things as passing under the eaves of houses, you still get wet. When you are resolved from the beginning, you will not be perplexed, though you will still get the same soaking. This understanding extends to everything."
And carry an umbrella.