Sleeping on the right side allows your spine to take it's natural position which is thus more comfortable, however sleeping on the left (which may also straighten your spine, it was unclear) can constrict your organs depending on their placement, but allow greater blood flow. The left side is also recommended for pregnancy, and sleeping with a pillow between your legs on either side is recommended. Sleeping on your back will take stress off of your organs organs and lungs but can weigh down other areas, and sleeping on your stomach can help with acid reflux.
More major organs do appear on the figure's left side (the pancreas, kidney and spleen, along with the heart) so I can see how it could be seen as constricting by sleeping on the left side, but I don't understand how it would help blood flow.
Pillow between legs and on side helping the spine.
Pillow reducing weight on pelvis under the knees, and one below the thoracic curve. Better pillow for neck support.
Pillow beneath pelvis to take weight off lower back. Better pillow for neck support.
Surveys show that people tend to sleep on their sides more than any other position, but it doesn't state which side. From the everything I've read and the diagrams above, I would assume that the right side is the most beneficial.