The hyperbolic quadrature method of moments (HyQMOM) for a one-dimensional (1-D) kinetic equation (u in R$) was introduced in prior work. This method provides a moment closure for the free-transport term of the kinetic equation that leads to a globally hyperbolic system of moment equations at any order of closure. Moreover, it provides the complete set of eigenvalues for the spatial flux that can be used in a Riemann solver such as HLL. Because the closure is based on the theory of 1-D orthogonal polynomials on R and their relationship to 1-D moment systems, the extension of HyQMOM to 2-D and 3-D systems is challenging. However, by using the generalized quadrature method of moments (GQMOM), all moments of the 1-D distribution functions are known. Here, HyQMOM is extended to multidimensional kinetic equations using a Grad-like expansion. As in 1-D, the multidimensional HyQMOM closure is defined based on the properties of the orthogonal polynomials Q_n (x) on R that are uniquely defined by the set of the 1-D velocity moments, including the moments of arbitrary integer order found with Gaussian-GQMOM. The characteristic polynomials in each coordinate direction are employed to define a 1-D polynomial basis. The direct product of these polynomials then defines the multidimensional HyQMOM closure using a Grad-like expansion. Focus is on closure of the free-transport terms in the 2-D moment system, and the range of hyperbolicity and the realizability of the latter for weakly collisional systems.