If it were my house, I'd call a local asbestos abatement company to come in and test what remains. You might not have anything to worry about, but if it's there, they can best advise you on how to proceed.

For what it's worth, I wouldn't be concerned about asbestos underneath a new floor. It's what is released when the material is disturbed that is the problem, from what I understand.

Go forth and sin no more. Chipping up asbestos tile is no big deal. Just cut them out, chip 'em up and bag them. The ONLY time it is an issue is if you grind them up into "fragible" parts where they are released into the atmosphere as dust. Even then, the only problem is (apparently) workers that have done the "grinding" for YEARS. Trust this; You're good to go.

The people who get cancer from asbestos are:

1 - Heavy smokers
AND
2 - Have extremely high exposure to air-borne asbestos for a period of several years.

This means shipyard or construction workers spraying it onto surfaces, miners, processors and the workers who remove the "friable" (crumbling) stuff from around old beams and boilers.

I've ripped out several vinyl tile floors and even scraped popcorn off ceilings ... and my lungs are fine.