Well about a dozen vinters have been doing so, some bodegas for hundreds of years now. The area known as La Geria is unique unlike any vineyard region in the world. This area is a proected area because of how the landscape has been sculpted by this process. There are hillside after hillside with zocos - low, crescent shaped dry stone walls made of volcanic rocks. These zocos each protect ONE vine from the elements. According to some by summer it looks "like emeralds lying in a black velvet case". Unfortunately we were there during the off season and so all the vines were without any color. But the zocos looked very cool going up and down the mountainside.
El Grifo is the oldest vineyard on the island, running continuously since 1775
Some zocos with some off season vines
Cool volcano we saw as we were driving thru La Geria
Zocos running up towards the top of the volcano
We bought a few bottles from this bodega as we'd had some of their wine with a few of our meals and really enjoyed it.
In addition to the zocos, the picon (volcanic gravel) traps the little bit of moisture they get on this island which nourishes the vines enough to grow. There is no insect disease here either which enables them to plant the stalks directly without grafting them (I know nothing about this process).
And to tell you the truth, it's not half bad. I'm no wine expert but I thought the wine I had here was delicious, smooth and crisp. We were told the reds aren't that great and to stick with the whites and that's what we did even though we aren't usually white wine drinkers.