We report structural, magnetic, dielectric and thermal properties of single-crystal BaMnO(2.99) and its derivatives BaMn(0.97)Li(0.03)O(3) and Ba(0.97)K(0.03)MnO(3). The hexagonal 15R-BaMnO(2.99) perovskite phase is a known antiferromagnetic insulator that orders at a Néel temperature T(N) = 220 K. We find dilute Li and K doping change the ratio of cubic to hexagonal layers and cause drastic changes in the dielectric and magnetic properties. Unusually large high-temperature magnetoelectric shifts (up to 85%) are observed near temperatures at which pronounced peaks in the dielectric constant are observed for applied electric fields along either the c or a axis, respectively. The temperatures of the dielectric peaks are strongly correlated with anomalies in the c- or a-axis magnetic susceptibility and the specific heat for all compositions studied. All our data suggest that the strongly anisotropic magnetic and dielectric anomalies (which occur near, or above room temperature) originate from the same Mn ion sites, which implies these materials form an exceptional class of magnetoelectrics.