I just go with an ordinary compass used to draw circles. Probably best to get one that can have two leads in it, but sharp point are OK (they just leave little pinpricks in the map, and actually prevent the compass from slipping).

Put one point (the starter point) on your start location. Open up the compass so the other point goes to the first significant corner, then place that point on the map where the corner is. Now, swing the compass so the "starter point" is behind the direction of travel, but the compass is aligned with that new direction after the corner. Put the starter point down on the map, and open the compass up more for the next straight bit. Keep following this procedure until you have a sizeable opening on the compass, then measure it against the scale on the map. Record that intermediate distance, then restart the procedure for the next sector. It sounds a little more complex than it really is.

I've used it to plot randonnees and it is very accurate for moderate to large-scale maps (100:000 to 250:000). It is wonderfully accurate on street atlases. And cost is all of what? $3 at a stationery shop, and you don't need dial-up. broadband or Windows!