How about this

 furious[a_, b_] := Module[{a1, a2, a3, b1, b2, b3, c},
 {a1, a2, a3} = Transpose[a, {2, 3, 4, 1}];
 {b1, b2, b3} = Transpose[b, {2, 3, 4, 1}];
 c = {-a3 b2 + a2 b3, a3 b1 - a1 b3, -a2 b1 + a1 b2};
 Transpose[c, {4, 1, 2, 3}]]

Timing results (from march's answer) for version 10.4.1

 list1 = RandomReal[{-1, 1}, {32, 32, 32, 3}];
 list2 = RandomReal[{-1, 1}, {32, 32, 32, 3}];

 l1 = listCrossMarch1[list1, list2]; // RepeatedTiming // First
 l2 = listCrossMarch2[list1, list2]; // RepeatedTiming // First
 l3 = listCross[list1, list2]; // RepeatedTiming // First
 l4 = furious[list1, list2]; // RepeatedTiming // First
 
 2.67
 
 0.6064
 
 0.0386
 
 0.0015