Regarding your specific request about wood chess sets:
If you will be visiting the Alhambra in Granada, there is a shop on the grounds specializing in parquetry – Recuerdo de Granada, Artesania Granadina, Hecho a Mano, Miguel Laguna, Real de La Alahambra, 30 – it is near the parador.
They have two grades of goods, the more expensive uses bone (ivory) in the design(s). The goods run the gamut from coasters, trays to traveling chess sets (in various sizes), and larger items. The items are manufactured on or near the grounds. I don’t specifically recall tabletop chess sets but there were so many items offered for sale that I might just have overlooked/forgotten about them.
Talk to the staff in the shop and they will tell you how the items are made.
Perhaps with the above information you can obtain a local phone number in order to make inquiries.
You will also see examples of these goods at El Corte Ingles, though I think buying them at the Alhambra will only increase their keepsake value.