Start by drilling holes in the stump. Make the holes as large and deep and closely spaced as possible. Fill the holes with sugar, molasses, syrup, old jellies, jams, candies, buttermilk or fertilizer of any kind. This is a good place to dispose of any synthetic fertilizers you might still have around the place, although organics would certainly be better.
Next, wet the stump and additions and cover with a thick layer of shredded mulch. How long the rotting takes depends on the type of tree. Bois 'd arc will take the longest to rot using this or any other technique.
A more practical approach for some is to hire a tree care company to grind the stump out mechanically. It's quick, always successful and doesn't cost much.
Don't pour any toxic herbicides into the holes as is commonly recommended. The toxins will eventually seep out and contaminate the soil and the roots of nearby desirable plants. Plus, the toxic products are not needed. The food products work just as well or better.
Large Stumps - Best technique is to hire an arborist to use a stump grinder.
Second best choice - cut stump flush with the ground, drill holes - larger, deeper, and more the better. Fill holes with sugar, fertilizer or buttermilk. Moisten thoroughly and cover with a thick blanket of mulch such as shredded tree trimmings.
For small stumps, jam a tin can over the stump and pound into the ground.
Potassium nitrate (saltpeter) can be added to the holes to act as an oxidizer and speed up the rotting. Using 2,4-D or other chemical herbicides is a dangerous use of these disease-causing and soil contaminating toxins.
Mesquite trees stumps on ranches can be killed with a 50/50 mix of molasses and diesel. Not a pure organic approach but works well and the diesel breaks down quickly.