This work is concerned with approximating a trivariate function defined on a tensor-product domain via function evaluations. Combining tensorized Chebyshev interpolation with a Tucker decomposition of low multilinear rank yields function approximations that can be computed and stored very efficiently. The existing Chebfun3 algorithm [B. Hashemi and L. N. Trefethen, SIAM J. Sci. Comput., 39 (2017), pp. C341-C363] uses a similar format, but the construction of the approximation proceeds indirectly, via a so-called slice-Tucker decomposition. As a consequence, Chebfun3 sometimes unnecessarily uses many function evaluations and does not fully benefit from the potential of the Tucker decomposition to reduce, sometimes dramatically, the computational cost. We propose a novel algorithm Chebfun3F that utilizes univariate fibers instead of bivariate slices to construct the Tucker decomposition. Chebfun3F reduces the cost for the approximation in terms of the number of function evaluations for nearly all functions considered, typically by 75% and sometimes by over 98%.