The word of the day is placentophagy:
A Las Vegas woman has won the right to keep the placenta from the April 12 birth of her daughter. The woman planned on giving it to a friend to dry and put into capsules, believing it would help ward off postpartum depression. According to the article, there is no Nevada law that prohibits hospitals from returning the placenta to the mother, but officials at Sunrise Hospital and Medical Center initially refused to hand it over, saying it was a bio-hazard.
While it is probably more risky to eat a placenta as opposed to a hamburger, my guess is that the risk of getting sick is still pretty tiny (although just thinking about eating a placenta is making me a little nauseous). And while there is little evidence to suggest that, at least in humans, eating the organ is beneficial, I respect the right of any woman to keep her placenta and make it into a lasagna. Just be sure to warn me before you invite me to dinner.