Hops. The golden foliage ones are especially attractive.

Silver vine (Actinindia polygama)

Any true grape vine including the ornamental types like Vitis coignetiae and V. vinifera 'Purpurea'

Schisandra (Magnolia Vine)

Among the annual vines, just think of those you can eat: Scarlet Runner Beans, Cucumbers, heirloom vining Squash, and Nasturtiums.

I would check with the National Poison Control Center before I planted anything. They are very nice about answering pet related questions: 1-800-222-1222.

Thank you, never thought of these vines. Great help, I'll look these up. Its too bad, cause I love Boston Ivy but can't have it.