Similar to clematis, lilies enjoy having their heads in the sun and their feet in the shade. To help keep their roots cool and moist, many experts advise mulching. Shredded bark, pine needles, leaf mould, or composted manure are all good choices. I prefer to grow ground covers under my lilies, having had excellent success with sweet woodruff, bugle plant (Ajuga), and even thyme in truly sunny sites. Some annuals work well too, and standbys such as alyssum and lobelia often have the added bonus of self-seeding so you won't have to replant them again next year. (If you're going to go this route, make sure you don't choose any of the super-invasive ground covers such as mint or creeping jenny.)
In Fine Gardening (# 61, p. 12), Gary Keim suggests some other excellent choices to a gardener asking what she can plant under her lilies to help keep them cool and happy:
"Lady's mantle (Alchemilla mollis), Anemone X hybrida, and Eupatorium rugosum 'Chocolate' are wonderful perennials to plant at the lilies' feet for sites in full sun, and all are hardy to USDA Hardiness Zone 4. Shrubs such as Caryopteris 'Worcester Gold' (Zone 7) and Cornus stolonifera 'Silver and Gold' (Zone 2) also fit the bill in full sun."
"Gardening in part shade narrows the choices, but versatile shade-tolerant perennials like astilbe, snakeroot, and hosta provide a palette full of colors to match with lilies up to Zone 4. Hydrangea macrophylla cultivars (Zone 6) and Kerria japonica 'Variegata' (Zone 4) are perfect companion shrubs for lilies planted in part-shade gardens."
Keep in mind though, that lilies can become finicky if their root zone is too crowded; Joseph Hudak warns that lilies can "dwindle into minor importance if too close to the moisture-seeking root competition or shading from nearby shrubs and trees" (Gardening with Perennials, p. 204). I wouldn't plant them close to any large shrubs, if I were you, and daylilies are a much better choice for underplanting trees.
