Photo courtesy of benketaro at Flickr.com.
One purely organic way to amend a clay soil is to work in rice hulls, the fibrous material left over after rice processing. Though nearly devoid of nutrients (except for a trace of nitrogen), rice hulls will effectively lighten and loosen a heavy soil, adding organic matter that helps retain water and is slow to decompose. Because of its disadvantages as a fertilizer, it works best when used in concert with compost and other organic soil amendments.
You may be able to find it in some feed and gardening stores, packed in enormous bags due to its lightness; otherwise, it can be obtained from rice processing operations. You can also use coir, known as coconut fiber, the same way as rice hulls. Check out coco fiber growing medium, cocunut fiber pucks, and organic potting soil with coir!