If you know the rough form of the functional dependencies, try a multiple linear regression. You can even do that in Excel, and with only three independent variables you would only need a few parameters. Try regressing this:

F(x,y,z) = a0 + (a1 x log(x)) + (a2 x exp{a3/y}) + (a4 x exp{a5/z})
[download]

With any luck at all that will give you a reasonably good approximation while only fitting six parameters (a0..a5). YOu didn't indicate how *much* data you have, and if you need to interpolate or extrapolate, which are really important factors in selecting a method.

Other options include finding a multi-dimensional spline libraries (Matlab has one, I think) somewhere. Alternatively, Tilly's suggestion reminded me of the loess smoothers. Those work by considering a span of "nearby" data-points to estimate the local shape curve. There is a multi-dimensional implementation built into the R programming language. The major problem with loess is that memory usage is a quadratic function (O(n2)) of the number of data-points.

As for your specific function, you may find it worthwhile to do some transformations first. For instance if I understand your description, then log(F(x,y,z)) is roughly of the form K*log(log(x))/(y*z). So log(F(x,y,z))*y*z/log(log(x)) is roughly a constant.

This is good because the estimator that I provided is going to give the best results when approximating functions that are roughly constant. (Cubic splines, etc, give very good results at approximating functions that locally look like low-degree polynomials.) And estimating this "rough constant" gives you (after reversing the above calculation) your original function F.

In general a judicious application of general theory and specific knowledge about your situation is more effective than abstract theory by itself...