Though I'm not sure what processes would do that. Without microorganisms, nothing rots. Lightning would eventually burn down most forests, but maybe everything else would just lie where it fell for ever.
That is not (quite) true. The lifeless bodies of flora and fauna won't rot, but they will decay. Macro-molecules (like proteins), the basis for all life, are fragile and easy to break down due to a range of physical causes, most likely of which is just the non-cryogenic temperature. Smaller organic molecules are more sturdy and will last longer, but in the long term they'll end up under heavy layers of sediments and turn to fossil fuel (coal & oil).
Which is also what "turn to fossil fuel" implies.