This is simply done by not unlocking any watch points when finishing a level. Stick to the roads and go directly between Lumber Mills and Log Stations, and you will be fine on the Island Map. I've included a description below of a walkthrough which doesn't go near any watch points, and which also shows how to get 'Forester' and 'My Only Love' at the same time.
(If you want to get 'Forester', don't press select, not even to change a vehicle, so try not to get stuck etc., or you will have to open the map to switch vehicles. If you don't change vehicle, this is a good way to also get 'My Only Love' - use just one vehicle to finish a map. It is useful to select the one vehicle you want to use as the MIDDLE vehicle out of the three you can select for the map - you spawn into the middle one by default, so you don't need to switch or open the map.)
Select Island map - it is best to familiarise yourself with it first, so start a new game and just use the Jeep to check it out and find all the Log stations / Lumber mills / Fuel stations etc., doing a practice run of the directions below, and then restart the level to do this achievement. (Probably best to go to main menu then start a new game.) Getting this achievement this way (and the other two if you want) only takes about 45 minutes to an hour.
I used the C-375 truck, but any that can take the medium log cart is fine. Good off road capability is useful. I used the log carrier and medium log cart (easier than using the small trailer or the long cart, and still only needs two trips per mill), and included a spare wheel in case I got a bit of damage. I then went straight and left, through the first lumber mill, up a small hill and left at the junction on the concrete road, heading roughly North East, then over the big ford, over the little ford, and along and down the hill to the North East Log Station. Load up, and head back the exact same way to the Lumber Mill near the start.
Do this twice, then you will be low-ish on fuel, so head straight down the main concrete road in the middle of the map, through the block post and down to the fuel station in the South East. Fill up, which will then be enough to finish the mission in the C-375. Head left to the 2nd garage, and past it to the Log Station just to the north (go via the road to the left, and head round to the right, if you don't want to get stuck in the trees). Load up, and head West, across the big track (this bit is very boggy up a steep hill and across a bad road - be patient and use Low 2 or 1),
Go through the forest on a little track, over and across the main road you came down earlier (DON'T check the map if you want the 'Forester' achievement!), and down a little track to a ford, heading South West, and just follow the track - it gets a bit vague over some sandy bits, but if you've familiarised yourself with it, you should be fine. follow that track all the way to the South West Lumber Mill, unload, and repeat (no need to fuel up again, you'll have enough in the C-375. As soon as you have put 8pts into both Lumber Mills, the achievement will pop if you haven't unlocked any watch points.
If you haven't opened the map (not even once / to switch vehicles etc!), you will also get 'Forester'. If you haven't switched vehicles at all, then you will also get 'My Only Love'. Enjoy! :)