1) Photograph Connection
One of the best ways to tell the a birth story is to look for connections. Pay attention to where hands or body parts connect during labor, and then compose and capture those connections. Some examples may be a support person rubbing the birther's shoulders, or he baby lying on the mother's chest. You can also capture connection by photographing where limbs connect to the bed, ground, or other form of support. These shots bring detail and creative story telling to the birth photos.
2) Photograph Emotion
There is no shortage of emotion at births. Watch for emotion in the birther and the support persons throughout labor, but especially during and after the baby's arrival. When the birther is pushing, this is a good time to secure a spot in the room where you can capture the babies arrival, but also the emotions of the parent/s. I typically find the side of the bed opposite of the birthing partner is best, because it's easy to get all reactions from that standpoint.
3) Photograph Perspectives
One of the biggest benefits of birth photography is that the birther can look back and see the story through others' eyes. At the birth, use varying perspectives to tell the story. Step far away from the birther and get a shot peeking into the room. Get very close to the birther (if allowed) to get details. Get low to the ground and shoot up toward the birther. Get high above (use a stool or something sturdy in the room) and shoot down on the birther, telling the story from above.