I used to follow Harvey's protocol, which is just a highly publicized version of natural hygiene, and the point is to have liquids in advance of your meal. Then when you eat to refrain from any fluids until well after your meal. This ensures that you don't dilute stomach acid and other digestive enzymes so your digestion will be efficient.
So then the answer to your question "Are we not supposed to drink anything with a lunch or dinner meal?", is Yes, no drinks with meals.