Firstly, it should be said, that, with time, all plants will drop older leaves, as well as flowers and fruit if any. Therefore, the distinction between 'messy' and 'not messy' is somewhat vague at best. Of course, some plants do drop mostly in one season and some drop more evenly over the course of the year. Which of these involves less maintenance is somewhat a matter of circumstance and perspective.
Perhaps, Bougainvillea seems particularly messy to some because their fallen bracts are brightly colored and especially light-weight.
Though the description does not fully explain all parameters, such as available light, (Photos would be an immense help.) I will go out on a limb (no pun intended) and suggest several flowering vines that should work well in Los Angeles: Passionflowers (Passiflora spp.), Trumpet Vines (Bignoniaceae, various genera), and, following the previous post's lead, Blue Dawn Flower (Ipomoea indica). For something a bit more rarefied, hunt down a Cup-of-Gold Vine (Solandra maxima). This is by no means, a complete list.