Johan B.
Google
This forest is part of UNESCO world heritage and it's a beautiful hike. If you're coming from Nara Park, you can enter the trail from just east of the restaurant Mizuya Chaya. Going around the whole forest took us about four hours. It will take you to the Wakakusayama hilltop, the Uguisuno-taki waterfall, and then along the southern edge of the forest.
Some more detail: The full circuit is about four hours trek, mostly gravel path or asphalt, but quite a bit of incline in some places. Going from the trail entry to the hilltop takes about an hour, where you'll want to pause to enjoy the view, then another hour or so to get to the waterfall, and another hour and a half trek to the edge of the forest in the southwest. It's possible to meet cars along the way, but we never did, only two mopeds and two bikes along the full circuit. We met a handful of other hikers as well, mostly Japanese.
It's perfect to spend a morning or afternoon. If it's a hot day, don't be discouraged, as the path is almost completely in shade from the trees and the forest is pleasantly cool even on a hot day.
I recommend taking a lot of water along, as we did not see any water taps along the way. There are two restrooms we went by, one on the hilltop (where there are also vending machines with drinks), and one at the reststop on the southeast edge of the forest.