That's actually a very cool idea. How would you handle multiclassed characters? (Would a barb 1/wizard 10 get to roll 3d12 on that cure serious wounds?)
There are a couple of solutions.
A. Assuming the barb 1 is first level, they could heal 1d12+2d(4/6)s. If it had been wiz 10, barb 1 - that would be 3d(4/6). If it had been wiz 1, barb 1, wiz 2-10; then 1d(4/6)+1d12+1d(4/6). A bit silly but might work.
B. What I did in my homebrew system is class HD, but most levels in a class decides what that is - so a barb1,wiz10 has a wiz HD so 3d(4/6)s.
C. Gestalt the HD - class with the highest HD is the HD you have - so yes 3d12.
D. HD is whatever you took at first level. 3d12 if barbarian at first. 3d(4/6) if wizard at first.
E. HD is whatever you took last. 3d12 if barbarian latest.
I do like this idea. In my system I also added +CON instead of +level. Then again I played around with the entire equation so I don't think it really matters. Yes it will be more HP but if you have a d12 you already have a larger pool so you probably need it, so a barb 1, wiz 10 might then have a reason to exist in order to pick up d12s for healing (per the C,D,E options).
Shouldn't mess things around too much. Makes me wonder why WotC never actually tried this in 3.5, or paizo for Pathfinder. I guess in theory it is supposed to be fixed like fireballs are always d6s, but that doesn't account for martial classes having more HP and a harder time recovering them.