You can route a PCB using a CNC machine.
You could also route it by hand with a dremel tool and some patience - but it will be ugly, and the by-hand process isn't suitable for fine-pitch components.
Is the concern over the need for corrosive chemicals itself, or because of ferric chloride in particular? Because there are other means than ferric chloride (which is pretty nasty stuff), but they all require some form of corrosive chemical to work.
One method which I have seen used to make parts out of thin metal plate, and could possibly be used for etching a circuit board, is electrolysis. The only issue is that all areas that need to be removed must in some way stay connected during the process. I'm not sure how well it would work for PCBs, but I've seen the results for sheet metal parts in which the component layout was transferred to the metal via the toner-transfer method, so something like that may work. Copper sulfate would be needed, though (which is poisonous, but fairly safe when handled properly - just don't eat it - its sold as root killer, btw). You'd also need a hefty power supply.