Firstly I suspect that finding yourself waking up on your back is NOT causing your baby any harm. As the others say, you probably just can't feel him/her in certain positions. The hospital tests are really reassuring.
The advice about not sleeping on your back is however sensible and based on anatomy and physiology....
The heart pumps blood around your body. Blood circulates from your heart around your body in a series of tubes (arteries and veins). The aorta is the large artery which delivers blood full of oxygen from the heart to your body and therefore your placenta and baby. Blood is returned to your heart in another tube (a large vein) called the Vena Cava.
These big blood vessels run from your heart, in the the middle of your chest down your back to just below your belly button where they divide into smaller tubes. Your womb and baby lie in front of these blood vessels and when lying on your back their combined weight can squash these tubes and affect the way blood flows around the body. A bit like standing on a hose pipe.
This can make you feel faint, dizzy and sick as there is less blood flowing to your head. If you feel like this then there is also less blood flowing to your placenta and baby. Lying on your side takes the pressure of your womb and baby off these blood vessels.
Not all women feel unwell when lying on their backs, however, the further on in pregnancy you are, and the bigger your womb and baby, the more likely you are to be affected.
Sorry Mrfreeze, propping your head on pillows will make no difference as this does not relieve the pressure on the blood vessels which are situated much lower down the body.