I have my floor ripped up right now, with new stick down tiles going in soon. Its a bit of a pain to take up but here is what I did.
Mine ran right under all my cabinets...ect. I left most of this as it really never seen. I cut with a knife around the perimeter and the grabbed one corner and took the whole piece out. Mine was glued down foam backed carpet. I had lots of foam stuck to the floor. I used a combination of scrapping with a putty knife, wire brushing and constant cursing to remove the foam.
I planned to add a new layer of plywood underlayment but my floor was in much better shape than I figured. Once my painting
is done, the new tiles will go right on the old sub floor.