First, some quote from British Fluoridation Society webpage (which text is based on some article authored by dr Levine, author of "The action of fluoride in caries prevention. A review of current concepts"):
The relationship between fluoride and tooth decay is complex and probably not yet fully understood. However, it is known that fluoride interferes with the process of tooth decay in at least four ways: (1) If children ingest sufficient fluoride (...) up to 7 years of age the fluoride alters the structure of the developing enamel making it more resistant to acid attack. (2) (...) encourages remineralisation and ensures that the enamel crystals that are laid down are of improved quality. (...) low levels of fluoride in the mouth gradually improve the strength of the tooth enamel and its ability to resist acid attack. (3) (...) reducing the ability of the plaque bacteria to produce acid. This is a major factor in the prevention of tooth decay. (4) (...) if sufficient fluoride is ingested during childhood when the teeth are developing, it affects the depth of the fissures (grooves) on the biting surfaces of the teeth (...) thus reducing the ability of plaque to remain undisturbed.
Now, let's analyze what is written above. Points (1) and (4) are important when you're a kid, they doesn't matter to you now. Point (2) is about improved self-repair and acid resistance. This is some very good property of fluoride, but it may not be so well understood and so effective as it is often presented. Quote from "Current concepts on the theories of the mechanism of action of fluoride." (1999):
Comparative studies of fluoride efficacy have shown that higher concentrations in solution are needed in pH-cycling studies of dentine than in enamel to maintain the mineral balance or to induce remineralization. (...) future perspectives for fluoride applications should be found in the retention and slow release of fluoride after various combinations of fluoride treatment, the combination of fluoride and anti-microbial treatment, the individualization of caries prevention, and the combination of preventive schemes with new developments in caries diagnosis.
(So this is not so simple just to have low levels of fluoride.)
If you sincerely keep good dental habits which include also proper diet and mouth washing, then you could rely on your natural remineralization. Keeping/restoring proper pH level would be important here.
Point (3) is about bacteria and here things can get tricky. Bacteria evolve and it is not impossible that they will learn to deal with fluoride. Quote from "New insight on the response of bacteria to fluoride." (2012):
However, the precise effects that fluoride has on bacteria and the mechanisms that bacteria use to overcome fluoride toxicity have largely remained unexplored. Recently, my laboratory reported the discovery of biological systems that bacteria use to sense fluoride and reduce fluoride toxicity.
Actually, some really bad-ass fluoride-resistant bacterias are known to exist from a quite a long time now.
If you're not to use fluoride then again a lot will depend on your dental habits. If proper pH levels are maintained then "good" and "bad" bacteries in your mouth will keep themselves in some form of balance (hopefully).
Now, I think that it is important to explain what is the main function of toothpaste and general toothbrushing: it's mechanical surface abrasion. This is nicely summarized here:
Toothpastes contain mild abrasives which physically scrub away the plaque and food debris without damaging the tooth enamel.
This, in essence, has nothing to do with fluoride.
So, in my opinion, you can use fluoride-free toothpaste. If you'll be careful with pH levels, do mouthwash, be careful what you eat, then your natural tooth decay prevention mechanisms should work. But I don't guarantee anything.
Also, if I was to use fluoride-free toothpaste, I would like know if there exist some alternative chemical substance of similar properties. I haven't checked that and this is good material for another question.