Hi,
I suspect that the material is either 1/8" masonite or thin plywood. Either of which can be cut with a utility knife. You will have to make multiple passes and there isn't much room, but with a new blade or two you should be pulling out the DW in no time.
Casey

A pullsaw is what you want. Companies that sell woodworking tools have several designs---Rockler's is one company. The saw works by pulling it towards the ouser instead of pushing away. That means the blades are very thin and can get in narrow spaces.

Make a cut on either side of the DW and the piece comes right out.

thanks for the suggestions...

We wound up using our power drill and several drill bits to drill multiple holes in a line along the kick plate and then snapped it off. Now if I can only get the dishwasher delivered!

Same issue as original poster, but wondering (a) whether a utility knife will effectively cut the toe kick (a thin plywood), and (b) whether the toe kick can be cut on the sides of the dishwasher only and reused (by glueing back or reconnecting in some way where the cut lines are not terribly noticeable).

Yes, Jobadee, you can reuse the toe kick. You will do better with a flush cut saw than with a utility knife. You can get a flush cut saw at Home Depot etc. for around $10.

Here is a link that might be useful: flush cut saw at Woodcraft

Thanks for your help!