I kind of touched on this when I wrote about Liwa, but if you really want to get a feel for what the UAE and Abu Dhabi specifically were like, you need to get away from the cities.
Near Liwa, there is an annual Camel Beauty Contest right around Christmas. The purpose of this is to promote and preserve the culture. One of the criteria is "camel toe cleavage." I haven't made it out there the past two years, but hope to go this year.
Also, out in the desert near Liwa you can sometimes see guys taking out their falcons for a fly. I haven't ever been asked to try one out yet, but if you ask nicely, I'm sure they'll let you take one out for a spin.
Also, I'd like to mention the Abu Dhabi Natural History group. Shortly after we first got here we attended an event put on by them. We saw a video shown regarding the pearl diving history of Abu Dhabi. Something that we hadn't signed up for but something that I would have loved had we been equipped for was the camping on the beach followed by the pearl diving expedition the following day. This group has lectures every month with occasional camping trips or events.
Even though the cities of Abu Dhabi seem to be trying to shake the image of a nomadic culture by building as many modern structures they can, it's nice to know that there are some people that maintain the old ways.