I had this exact same problem when I bought my C last year. Got an alignment done, it was way out, but no noticeable improvement.
Swapped the front tires with one of the dual pairs, and suddenly the vibration was in the rear end. Aha! Tread separation on one of the former-front tires 50 miles later, put the spare on, and the vibration is down to about 20% of what it was originally.
I now have 6 new tires on order.
The tires all had good tread and were only 3-4 years old, but I suspect the alignment was bad enough to damage the front tires in some way. Or maybe the previous owner liked hopping curbs. Not wanting to risk it, I'm putting new tires (Bridgestone Duravis R500HD) on all around, and taking the remaining four 'good' ones (the ones originally on the rear duals) and giving them to my neighbor for his beater truck.