I would be against it.

Funding: I've yet to see any location where all roads are adequately maintained. Where will the money come from to build and maintain a whole new infrastructure?

Not to mention bridges. If cyclists use existing bridges it wouldn't be a totally separate system.

Land Use: Here in rural Florida a system like this would require bulldozing a whole new road system through the forest, on private property, a national forest, a wildlife refuge and state parks.
Ain't gonna' happen.

Cager Confusion: Many of them don't know what to do now when approaching or being approached by a bicycle. More separation will only add to this confusion.