This is my name for a metaphysical theory (yes, I said the “T-word” :) ) that was initially proposed by John Michael Greer a few years ago in e-mail conversations on the ADF e-lists. He developed it a bit, into an article that was published in PanGaia #43, under the (IMO somewhat confusing) name of “Eco-Paganism”. However, he has since turned his attention mostly to his sustainability work, and has given me free reign to develop and write about the theory as I see fit. I think that my interpretation is evolving along a slightly different track than JMG’s, based on the evidence of his article, but at this point both versions would still be clearly recognizable as the same basic idea.
NOTE: all of what follows is based entirely on a layman’s understanding of all the fields of study involved; I am not an expert in any of these areas. In the reading I’m doing I begin to see that I am not exactly alone in any of these ideas :), but I think that this, along with JMG’s article, may be the first attempt to fit the questions into a specifically polytheistic context.
As I (currently) conceive it, natural systems theology is fundamentally a synthesis of certain aspects of Whitehead/Hartshorne-style process theology (minus the inherent Christian bias of Hartshorne’s own approach) with systems theory.
Let’s break that down a little.
Systems theory basically says that everything can be understood as a product of the interrelation of its component parts, both with each other and with their environment.  The internal interrelation of the components define the system (rock, body, tree, society, biosphere) ; the system’s interaction with its environment is what produces change in the system. To take a very simple example: a rock is a system defined by the interrelation of its molecules. Interaction with its environment – being tumbled in a river, scoured by wind-blown particles, what have you – produces changes in the system, but in most cases the rock system is still recognizable as “a rock” (except in extreme cases, such as being disintegrated by a hammer or melting into lava).
Likewise with everything else. I am a system – my integral components undergo change (cells die and are replaced every day), but absent a catastrophic event the rate of change is sufficiently slow that I am recognizable as the same “person” for decades. And so on.
Systems theory further postulates that systems of a sufficient organizational complexity can attain consciousness.  However, as far as I can tell, pure systems theory stops at the point of making that statement, and has been faulted as being insufficient because it does not identify the causes of this shift. A number of movements and theories have been proposed to fill that perceived gap; I suspect this may be one of those areas that can’t be clearly defined. Is a rock “alive”? Maybe… Is a tree alive? No question. Does it possess consciousness, or even self-awareness? Some would say yes; I’m not that certain, but I don’t deny the possibility out of hand.
What about a river? It’s a fairly highly complex system – one of its primary components, water, is in a constant and rapid state of change, and at least some of its other components (vegetation, water-based life forms) are highly complex as well. In fact at least one component, fish, itself possesses some level of consciousness. Does a river possess consciousness? Science and the ordinary-reality mind would say no; and yet there are numerous examples from across the globe of cultures that quite firmly believe that trees, rivers, springs, stones, mountains and pretty much everything else in the natural world either possesses consciousness or has a spirit associated with it in some way – from Shinto kami and Greco-Roman nymphs to the animistic nature spirits of indigenous cultures on every continent.
This is where I bring process theology into it.
 At which point it blends into complexity theory; where those boundaries are, if indeed they are firm at all, is outside my knowledge at present. See http://www.blackstarreview.com/rev-0120.html for a very brief introductory discussion.