If you have a very nice lawn, then you are going to be rather disappointed if you do not remove the stump and all of the main roots that branch out below the lawn.
Mushrooms are the scavengers of the plant world. They cannot grow unless there is dead and decaying material around for them to scavenge for growth and energy materials. If you allow the stump and roots to rot in your lawn, you will get to observe this process first hand on a large scale beginning 1-3 years after you cut down the tree and lasting for a decade or more after the mushrooms start to appear.
Your lawnmower will not cut low enough to remove them. And some of the mushroom varieties are as hard as wood and they leave a large divot in the lawn when you pull them out.
It is common to notice a circle of mushrooms in some guys lawn, along with several lines of mushrooms branching out radially from that circle. These mushrooms define the location of a former tree and its main branches and they will continue to do so for many years.
If you are concerned about your lawn, then: 1) Grind the stump down as far as reasonable, but at least 8". 2) "Chase" the main roots with the grinder and grind them out completely, don't just shave off the top of them. 3) Remove as much of the chipped wood as possible from the dug up areas. Raking is a minimal step, but screening the soil/chip mix is better. Very fine bits of wood are ok, but significant chips will feed mushrooms. 4) Reseed the areas.
You need to consider how fussy you are about your lawn and how much effort and money you put into it each year. Then consider the cost of having the stump and roots removed versus the hassle of mushrooms for a decade or so.
I spent hundreds removing the stump and chasing the roots of a 70' maple that I had removed from my front yard several years ago. I'll do the same in the next year or so for the second 70' maple which I now want to remove. I consider it money well spent. Some of my neighbors think that is wasteful and they are content to live with their mushrooms or remove them weekly. None of us are "right or wrong", its just that we view the situation differently and have different priorities.
If you want to conserve money but still get rid of the stump and some of the roots very close to it, then I'd recommend the saltpeter (KNO3 - potassium nitrate) plus kerosene approach. Properly done, this is quite safe, especially these days with the deflagration (explosion) inhibitors that are blended with the saltpeter in commercially available "stump remover" chemicals bought at garden centers. There is a lot of information about this method available on the Internet. I believe that a few months of soaking are recommended to permit the KNO3 to soak down into the roots. I also believe that the fire should smolder for days as it works its way deep into the root zone.
Cheaper sources of KNO3 are around. Some fertilizers such as "K-Power" (Vicksburg Chemical Company) are basically pure saltpeter. The chemical analysis on the bag should be about 13-0-44, which is the ratios for pure KNO3. I'm moderately certain that these fertilizers also have the deflagration inhibitors in them.
Just be warned that if you make a lot of inquiries into purchasing KNO3, you run a modest risk on getting on a terror watchlist. If you mail order the KNO3, you will certainly get on a terror watchlist. If you make inquiries about KNO3 on certain newgroups, you will definitely get on the list.
Final caveat - follow the directions on the stump remover packages. Also do a bit of Google searching for additional comments on using KNO3 plus kerosene/diesel fuel. Personally, I've advised a number of people on this method but I've never used it myself.
Good luck, Gideon