The sad news is if weeds don't grow, than perennials most likely won't grow. You can try to amend the soil, but there isn't much that likes a walnut.
You have several factors. Allelopathy and Hydrojuglone in the soil. Here's a list of tolerant vines, ground covers and flowers

anemone, aster, astilbe ,bee balm, begonia, bittersweet
calendula ,clematis (virginsbower), coral bells ,
Virginia creeper, daffodil, Shasta daisy,daylily
evening primrose ,fern, geraniums, hardy goldenrod
grape, wild hollyhock ,hosta ,hyacinth, grape; oriental
iris, siberian ironweed, jack-in-the-pulpit, lamb's ear
liriopelobelia, may apple ,morning glory, mullein
phlox ,primrose, raspberry, black rose, wild rudbeckia
scilla, sedum, speedwell ,spiderwort ,St. John's wort
sunflower, trillium ,tulip ,violet, wisteria ,yarrow

Also here's a good sight to visit:
http://www.ext.vt.edu/pubs/nursery/430-021/430-021.html

You can have a lovely garden beneath a walnut tree....but you have to remember to water...however not any more so than under, say a maple.
I would be planting some hosta, pulomoneria, heuchera and perhaps wild flowers like violet and trillium.....depending on your exposure.
The old wives tale is nothing will grow under a black walnut...but that's just not so....perhaps not what you first try....but there's lots of things that do very well.
Linda C

As usual, Linda gets to the heart of the matter. This topic has become distorted into much more of an old wives tale than factual substance.

Yes, it is true that a FEW things will not grow well under a walnut tree because of the juglone exuded by the tree. However, this list is fairly short (most notably, tomatoes). In other words, the list of plants that will grow under a walnut tree is very long. As Linda noted, the larger problem has to do with lack of adequate moisture reserves when located beneath a tree -- most any tree.

IronBelly