The Vias Verdes are "green routes" through Spain. Former railway lines, the gravel tracks are traffic-free and suitable for cycling and walking. They provide a beautiful alternative to on-road cycling, ideal if you have young kids in tow.
Routes are pretty flat or at least nicely graded, even in hilly areas, because they were originally designed for trains. The network is not huge at present but there are plenty of 2-3 day excursions to be enjoyed on the existing Vias. This is a really fun way to explore Spain!
www.viasverdes.es - only in Spanish, I'm afraid, but the map etc. is easy enough to follow and the routes are generally well-signed once you are there
Updates
The Via Verde in Jaén is called the Via Verde de aceite (oil) and is 55km long from Jaén city to River Guadajoz.It links up with the Via Verde de la Subbética, the Cordoba route.
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The best time to go is mid-May when the wild flowers carpet the countryside.