Such is the story with old houses Kristi, and you do have a reasonable concern that the floor tiles contain asbestos. If it is really rough tearing it up layer by layer, you might want to consider taking up the whole subfloor in sections. This is a scary proposition though since I can't see your situation, and can't really be sure it is a subfloor that you have there (versus the first floor).
With respect to asbestos tiles. The amount of exposure you'll get from removing the kitchen tiles is probably not too hazardous. Most cases of asbestosis occurred in people with long term chronic exposure and not acute exposures. However you should still use caution when removing the tiles, and wear adequate masks. I recommend the organic vapor masks (Martian masks) you can get at HD for about 30-50 bucks I think. It won't be much fun, but the other option of cutting subfloor is kind of frightening, and should only be used if you absolutely can't get the stuff off, in which case you'll also end up moving cabinets, etc. In that case, you'll also end up cutting the tiles forcing even more of the junk into the air, so I would strongly advise against the cutting.
Using a heat gun doesn't sound like a good option either since there is probably too much stuff on top of the tiles that would result in some pretty noxious fumes/odors.
Keep plugging along, you'll get it removed. Get some real good scrapers and metal tools, and don't worry about hurting the floor too much, you'll be going over it with your tileboard.