Different FourierParameters in each dimension for discrete Fourier transform

Question

I have a query about the Fourier utility, that does discrete Fourier transforms. I am unable to find any information on whether different FourierParameters can be used in each dimension. This is a natural thing to do in wave theory where you have $\exp(i (k x-\omega t))$ representations. Is it possible for example to use FourierParameters -> {1, 1} for the first dimension and {1, -1} for subsequent dimensions?

Background information (if useful):

Specifically, I use FourierParameters -> {1, 1} for the time dimension and {1, -1} for two spatial dimensions. I wish to take a Fourier transform of a large (960 × 256 × 256) array where the dimensions are $(t,x,y)$. This works, but is slow.

Ftxy[h_] := Module[{Hxy}, Hxy = Fourier[#, FourierParameters -> {1, -1}] & /@ h;
  Transpose[Map[Fourier[#, FourierParameters -> {1, 1}] &,
    Transpose[Hxy, {3, 2, 1}], {2}], {3, 2, 1}]]

This also works, but is slower

Ftxy3[h_] := With[{n = Length[h]}, N[Exp[-2 π I Range[0, n - 1]/n]]*
   Fourier[Reverse[h], FourierParameters -> {1, -1}]]

The raw 3D Fourier[] is fast, but multiplying through by the complex exponential vector is the slowest part. I tried setting FourierParameters -> {{1, 1}, {1, -1}, {1, -1}}, but that is not accepted.