(2001 Vulcan, 123xxx miles)
After a two week long battle with this taurus's lack of heat I have finally triumphed. At first I thought that I had a blend door problem, but after tearing the radio out i found that sadly it was intact. Next I moved into the engine compartment to start troubleshooting the heater core.
I tried draining and flushing the whole cooling system, and flushing/backflushing the core. At the time, I thought I had gotten all the junk out of it but in the end it turned out that the vice grips I used to block the bypass were not blocking the flow at all.
Finally I sucked it up and went to autozone and bought a new core. Using the pull-back-the-dash method listed here on the forum I replaced the old core. Once I could take a look at it I realized how blocked the thing was, both the inlet and the outlet pipes were totally blocked shut.
I hooked everything back up and let the car get warm. Much to my chagrin, no heat. I ended up having to pull out the entire hose setup from the water pump to the core to the thermostat. It will need to be disconnected in four places (water pump, heater core inlet/outlet, and thermostat). Becareful to have something to catch the coolant under the car. There were two nuts holding the hoses in place, the one on the passenger side being extrememly hard to get to (be sure to unclip the harness in front of it or you'll never get it off), the other is out in the open near the center of the firewall. To make room for the setup to be pulled out, I removed the air cleaner assembly from the throttle body back; and the drainage pipe for the driverside windshield sill (I'm not sure that is what it is called, it is where the wiper motor is located). With those out of the way, you can CAREFULLY maneuver the whole hose setup out from behind the engine.
Once it was out I finally found the problem. The outlet hose was completely clogged up to basically where the bypass hose attached. I managed to clean the entire thing with a decent gauge (maybe 3-4 mm thick) copper wire and carb cleaner. After cleaning every inch of the hoses and metal piping (On a side note, when you pull the hoses off the metal tubing the paint is going to be peeling off, I made it a point to trim it back and hit it with some steel wool to get the rust off) I used a simply cheap plastic siphon to push water through the whole thing in every way possible until it ran clear. Be sure to catch that too.
Now take that same siphon, back to the core inlet/outlet and flush water through it to catch anything that might be left.
Thats it! Just reassemble everything in order, paying close attention to that mounting nut on the passenger side (use something magnetic). Make sure all hose clamps and connections are secure and top off coolant as needed! Hope this helps!