Tink,
Groundclear (old name Triox) will do the job for you. Mix 1 quart with 1 gallon of water in a plastic sprinkle can (NOT A SPRAYER) and soak the ground six inches either side of the fencerow twice a year and you will have a nice clear strip of bare ground a foot wide with your fence down the middle. If you still want to screen the view of the other side of the fence, stockade fencing (Use the highest grade, the cheap stuff will only last about five years). Bamboo fencing will not only not keep out the weeds, it will deteriorate in just a few years.