I have a non-mapping GPS too (the basic yellow Garmin eTrex). If you're following a pre-determined route then you don't need mapping. (You do need mapping, or a paper map, if you come across a closed road or need to find the nearest station etc).
I create a route using bikely.com placing one point where you'd have a routesheet instruction (i.e. every junction or place where you turn off your current road). You need to turn off the "auto follow road" option otherwise it adds hundreds of extra points along the road.
You then need to name each point individually with the instruction and there's a 6 character limit and they must be unique! So I use two digits for a unique number, a space, then 3 characters for the instruction (R, L, SO, SOX, KR, BL, E3, LiR, etc which correspond to Right, Left, Straight On, Straight On At Cross-roads, Keep Right, Bear Left (usually at a Y or triangle), Exit 3 of roundabout, Left Immediately Right).
It gets a bit tricky to get this converted to a route of routepoints (rather than a track of trackpoints plus waypoints) but it's possible within Garmin Mapsource.
An example would be this ride:
http://www.bikely.com/maps/bike-path/167493 which is my usual training route round the Surrey Hills.
Once uploaded you tell the GPS to follow the route and put it on the big arrow/compass navigation page and it tells you what to do. In 4km or so, "L", and so you ride on for 4km. The distance to junction counts down until you get there and the arrow should swing over to point left, you turn left, and it moves on to the next point telling you how far it is.
This isn't the only way to use a non-mapping GPS. Many people use a track filtered down to 500 points and play the follow the arrow game.
There are lots of other tips/tricks with this but if you try it out you'll get used to it. There are certain limitations (120 points per route) and you need to split circular routes up otherwise it will point you to the finish when starting, rather than the route out.
