Dave,

If you use @Frank Kampas 's Excellent Post on Kuhn-Tucker Minimization. You get a simpler answer for these equations. They are in the form of AlgebraicNumber expressions (which can be turned into plain root expressions with RootReduce[]. This may be a bit closer to the answer you want (for simple equations that can be analytically solved).

RootReduce[
  KKTReduce[
   3 l*w + 2 l*h + 2 w*h, {l*w*h == 81, l > 0, w > 0, h > 0}, {l, w, 
    h}]] // InputForm

to get

{2*h*l + 2*h*w + 3*l*w, {Root[-54 + #1^3 & , 1, 0] -> l, Root[-54 + #1^3 & , 1, 0] -> w, 
  Root[-729 + 4*#1^3 & , 1, 0] -> h, \[Lambda][1] -> Root[32 + 27*#1^3 & , 1, 0]}}

Frank may be able to give you more insight...

Regards,

Neil