Create incentives for biking, but don't outlaw cars.
I do think cars should be banned in downtown-downtown though or at least have extremely harsh reinforcement: there is no reason for the rich businessman to be driving his Porsche at 120 km/h down a back road near where everyone cycles.
In my experience it's not the "rich businessman" but the "boy racer" with his Audi or VW Golf that are the most common problem.
In this way, cars can use the streets if they are indeed going to that area, but cannot move through the areas as a means to avoid roads meant to carry vehicles across town.
The city that exemplifies this is Groningen, Netherlands, and the superblock concept in Barcelona.