Shin Splits

I recently started running. I've been running for about four weeks now. I even got new shoes.
There is, however, a very annoying problem. I've got anterior shin splints, and they don't seem to want to go away.
I've tried different running stiles, different strides, different landings, different knee motions, different breathing patterns, different running material, nothing helps.
Here some people will say "dude, you have to rest and they will go away..."
But dude, I did! I stayed off my legs for a week (ok, 5 days, I cheated a bit), but they still came back in full force. As soon as I took the first step at any speed over walking, they started creeping back until they were like little goblins banging on my tibea with sharp rocks. As soon as I stop, though, they go away. The pain is not pervasive in any way, but it really keeps me from running.
I also don't want to wait for too long (without running, to recuperate), for fear that I'll forget what motivated me to run in the first place, and the flame will die. I do want to run. I used to run when I was younger, and it's a great way to stay in shape. The problem is that it hurts like a motherfucker, and I have no idea how to stop it from doing so, short of not running.
Any suggestions are welcome.