Since you say that there is no sign of mechanical fastener, it does sound like the fiberboard was glued to the subfloor.
I see two options for you:
If the new dishwasher has leveling feet, and they're the lowest point by a fair margin, you might be able to use a circular saw to make a couple of saw kerfs from the front to the back of the fiberboard. Have these kerfs line up on either side of the feet with enough clearance to give you some wiggle room. Using your crow-bar, hammer, chisel, some elbow grease and sweat, remove the fiberboard from between the saw marks to make two dados (or tracks) for the feet to slide back through. This will minimize the amount you have to remove.
If you don't have the clearance to use the above method, you'll have to remove the whole piece of fiberboard. Using your circular saw, make multiple cuts through the fiberboard in a variety of directions. Once you've cut up the surface, use the tools + elbow grease + sweat method to get it up off the floor. The cuts should make it easier to remove chunks at a time, since any place that doesn't have glue directly under it should be able to pop off once it's been released from a nearby glued section.
A couple of notes:
- You will, of course, have to ensure that your depth of cut is just deep enough to get through the fiberboard without actually hitting the concrete. You'd probably want to leave is a scant mm above the concrete to avoid damaging your blade.
- If your blade has brazed on carbide teeth, it's possible (though I don't know the likelihood) that hitting the concrete could cause a tooth to shatter/pop off the blade and it would become a dangerous missile. Use appropriate caution! (Even if this is an extremely low likelihood, I feel better leaving a caution than not mentioning it.)
- You may consider using a diamond-edged concrete cutting blade so it doesn't matter if you hit the concrete. They're not terribly expensive where I live. It's an option.
- You may find that only the edges of the fiberboard were glued down in which case your job just got a whole lot easier!. Break up the edges and the middle will lift right up.
The flush cutting saw recommended by Kris would go a long way toward replacing the hammer & chisel/pry bar, however, I think you'd still need to cut the fiberboard from the top in order to be able to get that flush cutter to a newly glued down area. You can do it without investing in new
toys tools if necessary, but, of course, no one can deny the joy of a new toy tool purchase. ;)