The aim of the present work is the 3D extension of a general formalism to derive a staggered discretization for Lagrangian hydrodynamics on unstructured grids. The classical compatible discretization is used; namely, momentum equation is discretized using the fundamental concept of subcell forces. Specific internal energy equation is obtained using total energy conservation. The subcell force is derived by invoking the Galilean invariance and thermodynamic consistency. A general form of the subcell force is provided so that a cell entropy inequality is satisfied. The subcell force consists of a classical pressure term plus a tensorial viscous contribution proportional to the difference between the node velocity and the cell-centered velocity. This cell-centered velocity is an extra degree of freedom solved with a cell-centered approximate Riemann solver. The second law of thermodynamics is satisfied by construction of the local positive definite subcell tensor involved in the viscous term. A particular expression of this tensor is proposed. A more accurate extension of this discretization both in time and space is also provided using a piecewise linear reconstruction of the velocity field and a predictor-corrector time discretization. Numerical tests are presented in order to assess the efficiency of this approach in 3D. Sanity checks show that the 3D extension of the 2D approach reproduces 1D and 2D results. Finally, 3D problems such as Sedov, Noh, and Saltzman are simulated. Copyright © 2012 John Wiley & Sons, Ltd.