Although you are left-right symmetrical on the outside, your insides are not, especially the arrangement of your digestive system.  Gravity will affect it differently depending on which side you are on.  

What JimL says. From your description, I should say that the problem is in your stomach.

If it is really bad, you might ask a doctor to give your gut a feel. Lying on your right side the succus entericus (it sounds better than "chewed-up food and stomach secretions in your sotmach") would lie on your pyloric sphincter, which should not matter, but...

And air and other gases would accumulate in the cardiac portion, near the attachment of the oesophagus. Normally I would not expect that to cause indigestion, so next time you see your doctor, better ask his opinion. It might have to do with food pressing on an ulcer or some similarly sensitive trouble spot that needs attention. If you eat a lot of coarse material that also could complicate the picture.

Good luck,

Jon

I agree with JR - I'm more likely to experience heartburn if I lie on my left than on my right. I don't imagine you're put together differently so it might be worth getting looked at. If you have to have a barium meal, it's not a bundle of fun so ask the radiologist to let you see the images - it's the best way to enjoy it.