We propose a novel approach to computing lower bounds for box-constrained mixed-integer polynomial minimization problems. Instead of considering convex relaxations, as in most common approaches, we determine a separable underestimator of the polynomial objective function, which can then be minimized easily and quickly over the feasible set even without relaxing integrality. The main feature of our approach is the fast computation of a good separable underestimator; this is achieved by computing tight underestimators monomialwise after an appropriate shifting of the entire polynomial. If the total degree of the polynomial objective function is bounded, it suffices to consider finitely many monomials, the optimal underestimators can then be computed offline and hardcoded. We present an extensive experimental evaluation of our approach in the pure integer case. In particular, we compare our method with baron, couenne, gloptipoly, and scip. It turns out that the proposed branch-and-bound algorithm clearly outperforms all the other solvers when variable domains contain more than two values, while still being competitive in the binary case.