Most of my navigating is done in either mountainous terrain or in deep, entrenched canyons (where you need to keep track of the number of bends you have traversed) and positioning yourself by landmarks is pretty straightforward. I have rarely used a compass....I still carry one, because when you need one, you really need it.
Pretty much the same for me. I always carry a compass, and practice with it regularly, but don't often need it.
I think it is paramount to develop really good map reading skills. In mountainous terrain, if you have a decent topo map, and pay attention to the terrain as you move over it, you can usually tell pretty closely where you are. If you are adept at reading contours, you can often infer a great deal from subtle changes in slope as you walk along. You can do a lot with good map and the old Mark I eyeball.
In my experience a compass comes more into play in flatter terrain, or when visibility of landmarks is poor due to forest, brush, bad weather or darkness. I've rarely if ever had to do a resection of three landmarks to find my position. More often I've used the intersection of a bearing to a single landmark with a linear feature such as a ridge or creek (sometimes called a "handrail" or "line of position"). I've also used the "angle off" method to find a spot on a linear feature. When I've used pace counting it is usually just to get around an obstacle and get back on course.