Diesel and a match should do the trick.

"Diesel and a match should do the trick"

It won't! They'll just resprout from the roots, with a vengeance.

Don't like using weedkillers, but they are essential for this. Cut notches in the bark and apply concentrated herbicide; early autumn is the best time if I remember right (could be wrong on this!). Cut down the trees as soon as they're dead, before they decay enough to become brittle and dangerous.

Resin

So you have to wait until autumn. OK. I have started digging around some of the already cut stumps. They pretty much have roots the size of a baseball bat and thicker. Should I just leave these roots bare to the elements, and deal with them later. Or go ahead and treat the roots with some herbicide, and try to pull as much up as I can? Thanks for your help.

Applying systemic herbicide this late will probably just mostly waste herbicide. It would have been best not to have cut the trees and to have waited until late summer / early autumn (before leaf drop). A systemic herbicide (Triclopyr, Glyphosate, etc) should be applied to freshly cut trunks or, as Resin was describing, by using a frill or girdle technique. Also, as Resin indicated, don't use premixed herbicides diluted for use in lawns or for weeds.