Well the ones that will climb naturally and do it all on their own you already have covered - peas, pole and runner beans, and cucumbers. Most everything else I can think of requires your "help" to make it climb - tying up branches, weaving ends, etc.

Certain varieties of sweet potatoes would be a possibility but it would reduce production. A few squash and/or melon types will trellis with lots of help and supports for the fruit.

Dave

Thanks, I was thinking about melons. I'd probably need one that matures fairly early. I've heard you can't grow sweet potatoes where I'm at.

Anyone have a certain variety of climber that is their favorite?
I've seen some youtube videos where people grew their tomatoes on these 'walls'.. maybe I'll just plant more of them?

Spinach Malabar is a nice climbing spinach-like green. It's very easy to grow and looks awesome. It's not my favorite green, but I like it enough to grow it.

I like achocha. Very nice looking vine,fruits can be eaten raw (when young) or cooked.
http://www.about-garden.com/a/en/2507-cyclanthera-pedata-achocha/

I sugget also HYACINTH POLE BEANS, if it is not in your list.
This bean is twice as nice; It has beautiful light purple flowers grown on clusters that turn into beans, similar to snow peas in shape but purplish. You can use young beans just like snow peas in stir fry or just by themselves. Once it starts flowering, will continue till frost.
Hardened beans MUST be cooked since it is slightly toxic uncooked.
But fresh ones are not. I plant it primarily as flowering vine and also for fresh pods. Never bother with collecting dry beans off of it, except for seeds.