There are various forms of shin splints, there are some good sites too be found on the web.
There's one form, as adam mentioned, where the tendons and muscles are tight and they effectively 'yank' on the shin bone on every impact while running, which can lead to pain and even stress fractures in extreme cases. The pain you get from this type of shin splints is on the inside leg, on the actual big shin bone.

Another form of shin splints is called compartment syndrome, this is where the muscle along the shin (on the outside of the lower leg) becomes sore, and the pain progressively worse as you carry on.

The cause of both types are the same however, impact. If you're feeling pain, while walking even, there's some damage already, so I'd recomend rest for now. When you get back to running, make sure you have trainers with serious padding on the sole, made for running, or as adam said, run on a field. This will soften the impact.

If you've got pain by just walking, there are shock absorbing soles available which you can slide into any shoe, this can help soften impact also.