E.B. Stone Organics: The simple answer is that organic gardeners don't use synthetic fertilizers or pesticides on their plants. But gardening organically is so much more than what you don't do.

Monstera adansonii, one of the lacier of the dramatic "Swiss cheese plants." Thrives in medium to lower light. Let top 1/4 of the soil dry between waterings.

Tradescantia zebrina is a vining favorite, tumbling over the edge of a pot with vibrant purple stripes and undersides of leaves. Bright, indirect light. Evenly moist (but not saturated) soil.

English Ivy (Hedera helix) comes in many varieties for trailing indoor vines. Bright, indirect light and water when the top half of the pot feels dry. Prune to thin vines occasionally.

Philodendron 'Black Cardinal' would grow upright rather than trailing with the others, but its dark foliage makes quite a statement! Bright, indirect light, evenly moist (but not constantly wet).

Burro's Tail (Sedum morganianum) can trail to 3 feet long! It would prefer bright light, so might be an experiment among lower light plants. Also fairly fragile to the touch, so best where it's out of room traffic.