This route goes all the way from Kyoto to Nara via two bikepaths linked with a short section of regular road. Since the main bikepath follows two rivers, first the Katsura River and then the Kizu River, the route is almost perfectly flat. And because it goes between two of Japan's major tourist attractions, there are advantages to going in either direction. We'll assume you intend to start out at Kyoto, the more centrally located of the two cities and the one with the most access options. Although the Kyoto-Hachiman-Kizu Jitenshado bikepath (shown in blue) starts at Arashiyama, it also passes through the western side of central Kyoto, so you can get to it from anywhere in central Kyoto - the nearest train station is Katsura on the Hankyu line, six stops from Hankyu Kawaramachi and the central Shijo-Kawaramachi area.
The bikepath on the Nara end seems to have two names - the Nara Jitenshado and the more specific Nara-Nishinokyo-Ikaruga Jitenshado. This route (shown in green) stops virtually at the Deer Park in downtown Nara, not far from both the Keihan and JR Nara train stations.
The latter bikepath also goes to Horyuji, one of Nara's most famous temples; if you have the time, you can cycle to Horyuji and do some sightseeing before cycling back and into central Nara.
(Updated links)
For a detailed map in English of the entire route, see:
http://nara-cycling.com/map/nara-kyoto-english/pdf/map.pdf
For additional information about the area:
http://nara-cycling.com/map/nara-kyoto-english/pdf/jyouhou.pdf
Other links:
http://www.cyclekyoto.com/kyoto-to-nara
(general description of the route)
https://www.youtube.com/watch?v=zbjSFQAlPGI
(YouTube video of the main part of the bikepath, speeded-up in places)
And here are the YouTube search results for a search of the bikepath name in Japanese: