I'm not sure this is a good answer to your question, but maybe a thought anyway:
Would you be happy to just keep it trimmed? Ivy provides a food-source for birds in the winter, and is a habitat for many insects as it is evergreen... the greenest thing you can probably do here is just manage it (As annoying as that might be) unless it is destructive. In which case...... errr... maybe get an indestructible fence?!
I know thats not much help, but I think a lot of being green is to work WITH nature, not against it. (Cos ultimately, we will fail at the latter!)

Ivy is a noxious weed here in Australia, so I know where you're coming from greenguru. It's extremely hard to kill, and if you do trim it, make sure you get every single piece that comes off - it can sprout again from a single darn molecule it seems. Put it all in a dark plastic bag and sit it in the sun for a few weeks - that should kill it. Then pop it in a biodregadable bag in the bin, and re-use your black bag (we don't have biodegradable black bags here, might be different elsewhere).
If you do a Google search for "kill ivy recipe" it should come up with a few remedies to try. Here's one I found:
http://www.naturalgardeneraustin.com/information/recipes/killpoisonivy.h...

Although I understand Steve's point about ivy still being habitat, this plant eventually strangles all other surrounding plants and sucks any goodness out of your garden. It's also toxic to many animals and other plants, and is a killer next to anything it can climb.
Once the ivy's under control, you can replace it with a locally indigenous climber more suited, which will encourage an even broader range of animals, birds and insects, not just those hardy enough to survive on ivy.