Yamasaki et al. have suggested that LaMn4Al8 is a nearly antiferromagnetic metal with linear chains of Mn atoms. The Sommerfeld constant for LaMn4Al8 γ=265 mJ/mol K2 is comparable to values found in other 3d heavy fermions such as (YSc)Mn2 and β-Mn. We have shown that LaMn4Al8 displays antiferromagnetic order below 15 K. Here we present inelastic neutron scattering data on LaMn4Al8. The magnetic scattering in LaMn4Al8 has two components: a broad inelastic response centred near 30 meV and a narrower quasielastic response whose width and weight change with temperature. We attribute the latter to near-critical spin fluctuations.