What are the major non-pilot causes of GA accidents?
For a statistical overview of General Aviation accidents I suggest checking out the Joseph T. Nall Report prepared by the AOPA Air Safety Institute. Everything I'm citing below comes from the most recent report (the 23rd report, covering 2011). 

The leading causes of accidents in 2011 were:


Pilot Error - About 77%
This includes poor flight planning, weather encounters, and a host of other things that we could at least theoretically avoid if we actually did the things we were taught to do in training.
Mechanical or Maintenance related - About 13%
This includes engine failures, control system problems, landing gear failures, etc.

The largest chunk of mechanical failures are "powerplant failures", which account for 38% of the mechanical failures, or a bit under 5% of total accidents.

Everything Else - About 10%
This includes freak occurrences like a flight instructor "walking into a moving propeller" (I kid you not, it's in the report!). A large chunk of these are also engine failure for causes that could not be determined.




If we add all the engine/powerplant failures together (the explainable ones from "Mechanical" plus the unexplained ones from "Everything Else") they come out to be  11.5% of total accidents, and are statistically the number two cause of Non-Commercial General Aviation accidents after "Pilot Error".