Hey guys, thanks for the responses.
After I posted my initial question, I went back outside and looked over the situation more thoroughly. It still didn't look very clear to me how to remove the heater core by way of the passenger compartment, but upon closer inspection, it did look rather straightforward to remove it by way of the spare tire compartment. I don't have a/c, if that makes a difference.
One advantage of attacking the problem from the spare tire compartment, is that you end up having complete access to the blower motor and the internal air passages of the heater assembly. It gives you a great opportunity to clean all the crap out of there. The toughest part by far was getting the hoses off of the old heater core. The rubber was practically bonded to the metal inlet and outlet pipes. I had to cut some of the hose off. I just hope I haven't made them too short by doing so. I guess I'll find out tomorrow.
It's actually an easy job, when you know how to do it.