So we decided to take out the old fence. Actually - my neighbors (including this world-famous blogger) removed the old fence and volunteer cherry trees growing along the border, and we had the new fence put in.
In this picture, you can see that the old fence/hedge yielded many a valuable treasure, including a luxurious pair of muddy longjohns!
The new fence still has that "too new" look about it, so we have our work cut out for us this spring planting some trellis and greening up the place (with anything other than ivy!). I've got cardboard piled on the ground to choke out the grass on the other end. I'm thinking raspberry trellis + some kind of groundcover?