In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm.