Clematis would be very pretty, although I imagine you'd have to give them trellis' to climb on, unless your wooden fence had some sort of texture - enough for them to grab hold of. Hollyhocks would be pretty. Honeysuckle too. There are several others that would be very nice and I'm sure you'll get lots of other suggestions.

A variety would be nice. All of the above plus sunflowers, if you put in a trellis you could also have a climbing rose or two. Clematis like to have cool "feet" so you will need to have a low growing plant to shade them. You would need a trellis for the clematis, they wouldn't just hang on to a textured surface.

I love the ideas you've given me so far. I hadn't thought of using vines on the fence but it sounds like they would look great. We put a sealer on the fence so it would retain it's natural light color and darker flowers would show off against it nicely. I haven't had very good luck with honeysuckle or hollylocks but could give them another try. I'd really like something to grow in front of any vines and I love hydrangea but I wonder if they'd get too large.

Hydrangea will not fit into your "fairly narrow line" part, but there are varieties that get 6' easy. I've got one.

You could also do morning glories or asperagus. I like the hollyhocks idea. I'm always cautious of honeysuckle - whatever fence they're growing on must be very, very strong because a honeysuckle vine can be quite heavy, especially after a rain and during a good wind.