Measuring clearance height for dishwasher
The clearance is the space you have to insert the dishwasher...so that would be from your floor, to the underside of the counter's front lip. Nothing bigger can fit through that space.

New counter tops can help if you have to make the clearance taller than what's currently there, but that will introduce other issues (like having to raise the rest of the cabinets to match). 

Finally, do the installers know they are pulling out existing cabinets? Typically an installer just inserts it into an existing opening. You may have to bring in a carpenter or cabinetmaker as well.