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