I always got bad shin splints all season long when I trained for track. Some people are just genetically pre-disposed to get them. They can be brutal, just walking around really hurts. Ironically, they don't hurt so bad when you are running.
Running on hard surfaces is the big problem, IMO. Try to limit your distance work, and run on the softest possible surface. When running laps, stay on the grass at the inside of the track. If necessary, get the coach to expempt you from team distance workouts and go to the gym instead for aerobic work on a bike or stairmaster. Once you get the shin splints, they take a long time to go away so try to nip it in the bud if you can.
Ice your shins and calves after training - freeze styrofoam cups with water and peel the foam away - makes icing easier. Not sure if it helps, but the doc always made me do it.