CMII owner here. First I tried new Genelex Gold Lion 12AU7's and 12AX7's in all five slots. A little bit better, but not night and day. So then I got an Amperex Bugle Boy 12AU7 from Andy Bowman at Vintage Tube Services, and that also is better. But I'm now still hankering to try a vintage Mullard in the pre-amp position, as stereo5 did.
I also splurged on new Genelex Golden Lion KT-88s to see how they sounded in comparison to the KT-120s, and after a week the 120s are back in the sockets. More impact, more bass extension, and only very very slightly less sweet. No reason (now, I discover) to change these.
So, my experience to date: the stock JJ's are not such underachievers as they are sometimes made out to be. But with some expensive NOS small-signal tubes you can certainly improve and tweak the sound.