Good luck with that! What are you going to eat? Can't have fruits, most modern fruits are very high in fructose. Definitely nothing containing starch, which will immediately break down in your mouth. Meat? It has some sugar too (largely responsible for the barbecue taste).
Its not wise to assume that mechanical (and perhaps even chemical) teeth cleaning wasn't practiced by our ancestors every since they evolved hands.
Sure, reducing sugars goes a long way in keeping your mouth healthy. But we can't rely solely on that.
Bacteria (even good ones) will ultimately form biofilms that need to be disrupted, otherwise they calcify and become tartar, then start accumulating down the gums, until they are mostly shielded from the air and the 'bad' anaerobic bacteria proliferate. You then get periodontitis and eventually start losing bone and then teeth (something accepted by society as a consequence of growing old, when it's in fact a disease).