Turns out that it might. Two new studies, one in the Journal of Physiology and the other in the British Medical Journal, build on previous work that indicates that sleeping on your back in the 3rd trimester of your pregnancy may increase the risk of still birth.
From a biological point of view this risk actually makes sense. When you are pregnant and sleeping on your back, the womb falls onto the 2 major blood vessels in the belly: the aorta that brings blood from the heart to the rest of the lower body; and, the inferior vena cava that carries blood back to the heart. The pregnant womb by the 3rd trimester is large enough that when it sits on them it can compress them potentially affecting the flow of blood to the baby. In most pregnancies this may not be a significant issue but in some pregnancies this can be enough to put the baby at risk which is why if I see you in the emergency room for whatever reason and you are that far along in your pregnancy you are always put on your side or sitting up (if you can) and not left to lie flat on your back.
Since adjusting your sleeping habits can take time, the best counsel is to start to sleep on your side earlier on in your pregnancy so it doesn't feel weird. Sleep on your left side as that side has been shown to be the best. And if you wake up in the middle of the night having flipped over onto your back or on the right side, please don't freak out. That short period is unlikely to have caused any harm, just go back to sleep on your side perhaps with some pillows to support you in that position and as with any pregnancy, if the baby isn't moving the way it was or anything else doesn't seem right, please make certain to be seen.