I always find myself having a dilemma. Save the forests along the river (because the Lumbermill gives you an extra +1 commerce there) or save the non-river forests so that I can build farms and cottages along the river (to get the river commerce bonus in my "worked" tiles in the early game).
Right now I'm thinking that the latter is much more beneficial, but it doesn't hurt to do some of both, since it's quite a while before your population gets high enough that you really need to work all river tiles (assuming you're lucky enough to have a river going through your city).
Wodan