It's not an old wives' tale - there is medical reasoning behind it, but you won't do any damage by NOT lying on your left side. Being comfy is the main thing!
Some doctors specifically recommend that pregnant women sleep on the left side. Because your liver is on the right side of your abdomen, lying on your left side helps keep the uterus off that large organ. Sleeping on the left side also improves circulation to the heart and allows for the best blood flow to the fetus, uterus, and kidneys. Ask what your doctor recommends in most cases, lying on either side should do the trick and help take some pressure off your back.
But don't drive yourself crazy worrying that you might roll over onto your back during the night. Shifting positions is a natural part of sleeping that you can't control. Most likely, during the third trimester of your pregnancy, your body won't shift into the back-sleeping position anyway because it will be too uncomfortable.
If you do shift onto your back and the baby's weight presses on your inferior vena cava, the discomfort will probably wake you up. See what your doctor recommends about this; he or she may suggest that you use a pillow to keep yourself propped up on one side.
ETA - I misread the OP - the left side is definitely the best side to sleep on, so the opposite of what you've been told is true!!