Apparently, you can get away with anything-- rigid plotting, reckless brainstorming, even inspiration from random marriages of unrelated terms--can lead to good fiction provided you approach it organically. This is what I've run into as I've surveyed articles on writing.
What is this magical organic writing of which they speak? Searching the whole of the Web via Google for "organic approaches to fiction," I get exactly nothing. I've decided that writing without the use of synthetic herbicides is not the answer. From the contexts of the remarks, I've attempted to derive my own answer.
First, as the articles imply, a good story can come from almost anywhere, even random term matching. One of my favorite short stories, "Of Mist, and Grass, and Sand," emerged from a match (or mismatch) of terms in a class writing assignment. And many of the things I've written myself have come from mishearing or misreading what people say. The title Catcher in the Rye is a twist on the song "Coming Through the Rye."
But, whatever you start with, whether it is a mistake or an assignment or something that begins in experience, I think the first step in an organic approach is owning it. Whatever the themes ideas or terms are, you need to understand them in a deep way and commit to sharing what they mean to you.
It's fine (encouraged in fact) to take your own slant. Alternate meanings and personal connotations work fine.
Connecting terms, especially those artificially thrown together, seems to be another part of writing organically. It may take some experimentation to find the proper glue. But determining the relationships between terms or images or diverse characters and ideas is worth the effort. Thing of it as a koan that, on reflection, may never provide a "right" answer, but has the power to unlock wisdom and insight you already possess.
Finally, there's the matter of growing your story or characters organically. This, I think, means letting things develop in their own time and in their own way. Writers may be tempted to push characters around, provide lessons based on their own values, or be clever. When the work is based on real life, it is extremely difficult not to shape the fiction to reflect what really happened. An organic approach a different path, giving the characters and the story the freedom to become what it is intended to be and favoring authenticity over convenience, convention, or pleasing the crowd.
I find my conclusion ironic. Most of the articles focus on making aspects of story creation easier, but all these constructs, once the focus shifts to doing things organically, exacts a cost in terms of risk, discipline, and investment.