Put on a surgical mask for starters. Then get a utility knife and a pack of new blades (about $.50) Assuming the flooring is glued to a lauan or some 'soft' underlayment, cut the flooring in any way you can to find the seams of the underlayement. When you find the seams, cut the flooring at the seams. Then try using an ice chopper (yes, the kind that you use to chop ice outside in the winter), and pry-up the sheets of underlayment (with the flooring in place.) If that doesn't work, or if you don't have an ice chopper, try somethign like a coal shovel or a large pry-bar. Safety glasses would be good too (about $10.00, even less.) Watch out you don't damage your lower cabinets. Get the sheets outside and fold/bend them into manageable sizes (that the rubbish folks will take.) Use the knife to 'mark' your bend-points. DON'T use a saw EVEN with a mask! You'll heat up the asbestos and particlize it! That's where the problem lies.
If you need more info, e-mail me directly. My best to ya and hope this helps.