Bruce, I've had a play , some time ago with this and it seems feasable. Mine was only 2 channels and I did play with a MUX for the thermocouple amps , but I had other things to get on with, so only ever got to the point of proving (satisfactorily the principles ) rather than hard prototyping.
There are many options re thermocouples which are a lot cheaper and the thermo. amps include cold junction references making it pretty straight forward. I'm in a bit of a spin at the moment (time wise) but I see if I can drag out what I have to refresh the grey matter.