Could it be something as simple as poor core-conditioning?
That's what I thought when I read the subject line and I bet that's the problem - it's the reason for lower back pain in many non-runners too.
You don't even have to do a lot to help fix the problem (if that's the reason). 2-3 short sessions of abdominal exercises a week and you should be good.
Learn how to do a pelvic tilt, too. Keeeping your torso in line when you run will help prevent back pain. One of the best tips I ever read for running is to think of a string going through your head down the back of your neck to your tailbone. Then imagine someone is pulling gently on the string so your back straightens. This does a number of things for your running form but it also keeps you from pushing your stomach forward and sticking your butt back, which puts strain on your lower back.