The way I do it is to score the copper track twice with a sharp knife, then apply the soldering iron to the cut section (with a dob of solder to aid conduction). This weakens the glue and the piece of track comes off easily after 2 ~ 3 seconds.
You can buy a trace-cutter from Veroboard which is just a drill bit in a handle. I simply twist a drill bit in my fingers or use a drill press. The trace is cut at a hole that has no wire in it.
Or you can opt for donut boards, they're much more customizable, and you can fit in components snugly.
here's what mine looked like: **broken link removed**
I used the extra resistor/capacitor leads which are not needed, clipped them off and put 'em in a container. Then, use it at will, which makes it look like a PCB.
A very important tool: a pair of tweezers to bend the lining with precision.