French drains are basically underground drainage ditches---trenches filled with gravel or tile and topping them off with soil. They may be used by property owners to drain standing water in low areas, gardens and along buildings. Planting perennials over or near a French drain is a simple matter, providing you know its location and perimeter.
French Drains as Watersheds
French drains are draw water away from damp spots; if they do their job efficiently, there's no reason that the ground above them should be soggy except during "gully-washer" rainstorms. The area above and around the trench forms a three-dimensional watershed system. Streamside plants like cardinal flower, rose mallow and a wide variety of ferns grow well along streams and would flourish in an area that serves as a mini-watershed for a single property.
Since your drain is not constantly running, use native plants like coneflowers, coreopsis and other plants that are at home with temperatures and precipitation patterns in your area. Although the swale or depression over most French drains dries up in the summer, the water that runs under the surface keeps the ground moist most of the time, much as is the case in stream- and pond-side ground.
Perennials for Damp Sun
Some fortunate low areas are located in sunny areas. These can be planted with perennials that like moist but well-drained soil like irises, monarda and daylilies will soak up moisture and even demand more during hot summers. Native plants that can stand in water in the spring and tolerate dryness in summer include prairie favorites like asters, black-eyed Susan, bellflower and sages. Peonies, hibiscus and pinks can also thrive in moist soil as long as they get enough sunshine. Ornamental grasses like fountain grass and big bluestem soak up water in the spring and tolerate dry summers.
Perennials for Damp Shade
Shady swales require woodland plants like columbines, wild ginger, trillium and shooting star that are at home in the moist open shade of wooded areas. Violets, Jack-in-the-pulpit, lantana and trumpet creeper are native species that can brighten a shady moist area between shrubs like spice bush, indigo or false indigo.
With an hour or two of morning sunlight, bugbanes, buttercups and black cohosh will flower in elegant spikes. Many woodland plants, such as trillium and Jack-in-the-pulpit are protected species and may not be gathered "in the wild"; check your nursery's sources to be sure that the plants have been legally produced.