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