It does not get any less expensive or flexible than this ... a trellis of vines suspended by a swagged chain (which hangs on beefy hooks screwed into the posts. Twine drops at regular intervals from the chain and is attached to stakes in the ground by tying. Once the vine reaches the top, the twine is immaterial. You can get super quick and complete coverage with annual vines, but in order to make it permanent, perennial vines need to be established. (So this is possibly a work in progress for a couple of years. Probably bare during the first winter or two.) The height of the screen is controlled by where one places hooks for the chain attachment. Made it too high? Place some hooks at a lower elevation and lower the whole thing. Made it too low? Place some hooks higher up on the post. ... "Oh, crap ... posts not high enough!!" Think that one through before beginning. In the picture is moonflower vine. It took about 6 weeks from seed to complete coverage. In the evening come hundreds of huge, deliciously perfumed pure white flowers. Heavenly! The maintenance is easier, I think, than with a hedge. The thickness is self limiting (depending on the vine). The height is self limiting. I trim the vines off the post finials on a regular basis ... last year, I didn't. It's a different -- kind of spooky -- look. Depending on the vine, it will probably send runners out at the base. These could become "groundcover", be trimmed off, or redirected into the screen mass. Vines as a whole tend to be rambunctious. For a perennial vine, I would pick one that can be sheared and is somewhat controllable ... Clematis, Bignonia ... not Wisteria or Kudzu. Another positive feature that one might not guess that's different from a fence or hedge, this screen reacts to wind and undulates in waves. So that's pretty, too.

This post was edited by Yardvaark on Wed, Aug 6, 14 at 20:52

This is actually much different than the case we had the other day. I don't think you should be limited to bamboo at all. With solid sun, and a zone 7(?), you really have many choices for broadleaf evergreens or conifers that could meet your criteria. I mention it all the time, but it's because it's pretty easy to find Thuja occidentalis 'Degroots Spire'. This is a narrow arborvitae, make sure that the ones you purchase are 'single leader'. This is very important when you have such a narrow space. Thuja also grows about 1ft p/year and can handle sheering quite well.

There are at least two types of upright boxwood that would work. Boxwood makes one of the best hedges around, but does require some patience.

Buxus sempervirens 'Graham Blandy'
Buxus sempervirens 'Dee Runk'

That last two types are upright Juniper and Cypress. There are actually several different cultivars in these families that would work. I just listed two of the more common ones. Juniper is terrible with snow load and is susceptible to blight, so that would probably not be my first choice, although it could work, especially with full sun.
Juniperus scopulorum 'Skyrocket'
Cupressus sempervirens 'Swanes Golden'

Because all of these have a very narrow profile, I would not anticipate you ever having to get on the other side of them to sheer. This will be an advantage, as I suspect there won't be much room on that side and your neighbor probably won't appreciate you putting a ladder in his yard to trim them!