85-88 Thunderbirds
I wrote it from memory after recieving many questions about the process.
Email me with any questions.
I think the best route is to remove the Heater Box from the car. Others have suggested (and it's been done) to cut a hole in the firewall (from the engine compartment) and remove the core that way. That's a hack job if you ask me.
Before you start, unhook the battery. Drain the antifreeze and discharge the A/C (have the R-12 or R-134A recovered by a mechanic).
To remove the box, you'll have to unbolt the dash. There are three bolts along the top (one in the center, under that little plate with the slots in it and one under each of the speaker grilles). In addition, there is a bolt on each side down low (it may be under the kick panel, I'm not sure). I think there is also a support which bolts to the firewall under the heater core box, near the center of the car (but accessible from the passenger side). It's been almost two years since I did this, but it also seems like I had to loosen (or maybe remove) a bolt above the steering wheel. I remember having the Gauge Pod out, and that's where that bolt was.
As you pull the dash out, you'll need to unplug any wires and vacuum lines that you can. I only had to pull the dash out about 18" on the passenger side to have enough room to get the Box out.
In the engine compartment, unhook the hoses from the heater core. Also, unhook the lines to the A/C Evaporator. There is at least one bolt that holds the Box on the engine side of the firewall that needs to be removed as well.
Back inside the car, there are a couple bolts that hold the heater box to the firewall. Once those are removed, the heater box should be loose. Just wiggle it out. Be careful not to scratch anything as you slide the box out of the car (some of the edges are sharp).
Once out, just unbolt the top of the heater core and replace it. Be sure to clean out all the stuff that's fallen down in there over the years. Also remove the A/C core and clean out under it. Be sure the drain for the A/C is open, too.
One thing I didn't do, and wish I had: Check the vacuum motors (the little devices that actuate the various doors, there are three or four of them). If they don't hold vacuum, replace them.
After all that is done, you're ready to reinstall.
It will take about 8 hours. Just plan for it and take one step at a time.