Gretchen, LJ is right; this will be laborious. But, the job could be made easier, I think, in this way:
Score the linoleum into 12" wide strips & peel up all that you can, using a putty knife to scrape where needed. Primarily you're interested in removing the surface; there will be some paper backing left behind.
Using a pump-up garden sprayer, dampen the remaining paper backing with hot water (a little dish soap might help, too), let it soak, re-spray, etc. a few times until the backing is pretty soaked through. You'll be wetting the hardwood somewhat here, too, but if you are planning to refinish this is not near the problem as the gouging of the hardwood that would come from razor blade scraping.
After allowing the dampened backing to set for 20-30 minutes, try scraping it loose with a 3" or 4" putty knife.
I'm fairly sure this method would prove to be effective, but you might first try one area to see if it does. You may also need to adjust the wetting/soaking times to make the job go well. Good luck