Difference from liquid to solid items
Python 2.7 - 659 548 373 355 bytes
My first golf, but still very golfable.
The rotation thing didn't make it as easy as expected though.

import numpy as n
L,W,H,l,w,h=input();bx=n.zeros((L,W,H));s=0
q=(L*W*H)//(l*w*h);o=[[l,w,h],[l,h,w],[w,l,h],[w,h,l],[h,l,w],[h,w,l]];p=n.ndindex
for i in range(q):
 for a,b,c in p(L,W,H):
  for k in o:
   d=n.copy(bx)
   try:
    for x,y,z in p(k[0],k[1],k[2]):
     if d[a+x][b+y][c+z]:raise
     d[a+x][b+y][c+z]=1
    s+=1;bx=d
   except:pass
print q-s

But still a nice little brainteaser :D
**Edit: My colleague and I pondered a little more, and we found some more optimization potential
**Saved 18 byte thanks to CatsAreFluffy
Examples:
[6,6,6,1,5,6]
0

[3,4,3,2,2,2]
2