I want to fly a continuation flight plan (for example, the Kangaroo Routes: http://en.wikipedia.org/wiki/Kangaroo_Route), but cannot find a way to do so in the default FSX flight planner. Is it possible at all? Or do I need to use two separate flight plans?

In the flight planner select the beginning and end airports and then just drag the flight path to any waypoints you'd like to use along the way. I'm not sure if this is what you were asking or not :-/

I've tried that before, but it doesn't work; FSX treats it like a VOR or NDB. I want to have a flightplan with stopovers.

I don't think the actual pilots would use one giant route, they probably, from what I understand, do one route at a time through the FMS. Yes, in flight simulator it would be easier to do one big route, but there is no way other than treating each airport as a waypoint.

You cannot make a FS flight plans with stop-overs.

Simple as that.

Solution? Make multiple flight plans.