Let Sleeping Logs Lie
The Living Dead
If dead standing timber it isn't going to fall on your house or car, leave it. In a state of decay, the tree is still a great home for the living, providing shelter to a multitude of wildlife from microbes and fungus to birds of prey. As it slowly disintegrates, it will feed the soil beneath.
Eventually the old tree will just fall over, continue to rot and provide habitat for ground-dwelling creatures. Remember, encouraging biodiversity is part of what makes a PRFCT place: each inhabitant has a role in a nature-based system. Removing biomass from your property is removing the food that your landscape has provided for itself. (And it's better than anything you can buy in the store!)