I have found, somewhat suprisingly, that the PI (V5 if you have reverb, V4 otherwise) and power tubes seem to make a bigger tonal difference in Mark IIIs than the other preamp tubes, which is very different than in other amps.
I've sprung for some NOS RCA and Sylvania tubes for the front end but haven't really found them to be worth the money for the difference they make. They sound amazing in *other* amps, but in the IIIs it's "eh." However, I run Mullard CV4024s in the PIs and reverb slots and that makes a big difference in clarity.
I run Siemens NOS EL34s in the outer sockets, which are great, but am still finding 6L6s I like. I will probably end up with the new production =C='s.