Long-distance paths have acquired quite a following in the past few decades.  In the US you have ways like the Appalachian Trail, the Pacific Crest Trail and the Continental Divide Trail standing as the Big Three.  There are plenty of other long-distance paths to spend days upon days on, as well, such as the Arizona Trail, the Green Mountain Trail, and on and on.  Of course, the American examples pale in number when compared to the Old World.