I was in Tenerife for 3 weeks. I'm not sure how Las Palmas compares.... but I saw a lot of people riding on out of the was roads. I've been to a lot of countries in a lot of places and I can tell you that I would be very careful about picking where I went on a road bike. The drivers are nuts!! As for a mtn bike, that might be a very good option if you can find trails etc. Most of the hiking trails I explored were very lightly travelled but easily followed and the rangers at the stations in the national parks etc were very helpful.
Have fun and enjoy. I loved it there.
Living life like there's no tomorrow.
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2007 Look Dura Ace
2010 Custom Tonic cross with discs, SRAM
2012 Moots YBB 2 x 10 Shimano XTR
2014 Soma B-Side SS