Sounds an attractive idea. Just as long as the roof is fairly waterproof, as there is nothing worse than settling in to sit out a heavy shower and getting dripped on.
The roof of our little open fronted 'hut' in the garden is made of overlapping wooden slats (recycled) and is mainly rain resistant. The sides are trellis with plants growing up them .. roses, early honeysuckle and ivy mainly, with Clematis Montana over the roof. Needs keeping under control bit smells lovely.