What is a good and safe way to remove 40 year old vinyl floor tiles from my kitchen?

I have a small galley kitchen which I am updating. I have replaced the sink/counter/cabinets/fixtures/garbage disposal and painted the remaining cabinets. After I paint, I am going to want to get rid of these old, cheap, stained, scratched, vinyl tiles. I don't want to pay Lowe's $400-500 or whatever to do this. ACO sells a box of 20 self-adhesive tiles for $10. BUT, it says "do not lay these over vinyl flooring." How can I get the old ones off?