Much research in Lagrangian staggered-grid hydrodynamics (SGH) has focused on explicit viscosity models for adding dissipation to a calculation that has shocks. The explicit viscosity is commonly called “artificial viscosity”. Recently, researchers have developed hydrodynamic algorithms that incorporate approximate Riemann solutions on the dual grid [28,29,35,30,2,3]. This approach adds dissipation to the calculation via solving a Riemann-like problem. In this work, we follow the works of [28,29,35,30] and solve a multidirectional Riemann-like problem at the cell center. The Riemann-like solution at the cell center is used in the momentum and energy equations. The multidirectional Riemann-like problem used in this work differs from previous work in that it is an extension of the cell-centered hydrodynamics (CCH) nodal solution approach in [7]. Incorporating the multidirectional Riemann-like problem from [7] into SGH has merits such as the ability to resist mesh instabilities like hourglass null modes and chevron null modes. The approach is valid for complex multidimensional flows with strong shocks. Numerical details and test problems are presented