We provide a fresh perspective on the pseudogap physics for attractive fermions as described by the three-dimensional Hubbard model. The pseudogap in the single-particle spectral function, which occurs for temperatures above the critical temperature T-c of the superfluid transition, is often interpreted in terms of preformed uncondensed pairs. Here we show that the occurrence of pseudogap physics can be consistently understood as due to local excitations which lead to a splitting of the quasiparticle peak for sufficiently large interaction. This effect becomes prominent at intermediate and high temperatures when the quantum-mechanical hopping is incoherent. We demonstrate the absence of the conjectured pairing temperature below which pseudogap physics is expected to occur. Our results are based on approximating the physics of the three-dimensional Hubbard model by dynamical mean-field theory calculations and a momentum independent self-energy. Our predictions can be tested with ultracold atoms in optical lattices with currently available temperatures and spectroscopic techniques.