The following are plants that will grow underneath walnut trees

Lilacs
Daphne
Rose of Sharon
Boxwood
Viburnum
Clematis
Astilbe
Campanula(Bellflower)
Hosta
Daylilies
Monarda(Bee Balm)
Peonies
Phlox, summer flowering
Crocus
Hyacinth
Tulips