Organic quilting I would take to mean free form and spontaneous. So organic lines in quilting look hand drawn, not ruler straight. Organic shapes show a lot of variety, and while there is repetition and balance, there is definitely a value placed on the handmade / unique aspects over precision and machined perfection.
An example of organic quilting:
http://www.flickr.com/photos/12590460@N06/8658246759/
The free motion quilting here is freehanded, spontaneous, and frankly not to everyone's taste. However, I did it to represent rain against a window pane, which was my inspiration for the whole quilt.