Hmm, you specify two different limits; your title says organic, but then you say do don't like the petroleum products. Since this isn't food we're talking about, I think you should clarify what you mean by organic, since petroleum IS organic. And the silicon is organic, in so much as rocks are organic. Do you mean renewable? Do you mean non-dyed/unbleached? Or do you mean pre-industrial (i.e. no bright-orange plastic-looking fabrics)? An elephant skin tent with ivory poles is organic, but I don't think that's what you want. A down bag with silk fabric is organic, but will you tolerate bright synthetic dyes? East Asian cultures have for thousands of years dyed their silk clothing bright colors, which can look very similar to some of the petroleum products. So are you looking for Western Pioneer organic (muted colors, local fabrics), or can you still wear bright red so long as it is 'organic'? Further, some would argue using petroleum-based synthetic insulation is more humane than using goose down. Finally, are you looking for comercially available organic? That's harder to find, as it will always be more expensive, or are you willing to make/hunt down your own fabrics?
I'm sorry to play devil's advocate here, I'm just pointing out that your topic non-specific, and there are a vast number of local 'natural' products that vary widely amongst many cultures. It was the synthetic products that unified the global fabric world. Tell us what your philosopy is, and we can help you narrow down on the materials.
I'm only being mean like this because I have thought about going non-synthetic for years, and haven't been able to find suitable substitutes, particularly in the lightweight waterproof category.