This "natural" approach is not new, it was the only approach a hundred years ago. Gregory Bateman's Freshwater Aquaria
published in 1890 referenced a balance between animal and plant life. Although little did he and others at the time know how profund this was, since they did not posess the knowledge we have today about bacteria which is a significant factor in making this work.
MoneyMitch, you're right, I did mention in a thread a while back that I knew aquarists who maintained healthy tanks (they say) with no filter, no heater, one light, many plants, fewer fish (than most of us), and no partial water changes. As Mikaila31 has said, the method is sound, but once again it comes down to the balance. When everything is in balance, the aquarium functions naturally. Jeff Walmsley in an article in the December 2008 TFH talks about the balanced aquarium and at the conclusion mentions that his idea of a balanced aquarium has 1 small schooling fish for each 3 or 4 gallons of water. Most of us are not satisfied with a 60g aquarium that has 15 small fish [such as neon tetras] and that's it.
With fish stocking well above such a small level, as most of us have, the partial water change is mandatory to maintain the same balance that is the essential element in any healthy aquarium. So, nomel, even as Diana says that the plants utilize ammonia and nitrite to convert it to ammonium before it gets to be nitrate, you won't get out of those weekly partial water changes unless you have near-fishless tanks!
Thanks Mikaila31 for the link to Diana's article. I have picked up some of that info from other authors' articles, and I've frequently commented that plants in a new tank greatly assist with the cycling. But Diana puts it all together and makes it very clear, and I am glad to have come across her article. I must confess to not yet having her book, I've been putting it off not out of mistrust but simply because I'm perfectly satisfed with the results I have achieved over 12+ years and haven't seen the need to find another method. However, for the knowledge and insight it obviously contains, I do intend to get it. One never stops learning.