The tunneling anisotropic magnetoresistance (TAMR) effect demonstrates the sensitivity of spin-polarized electron transport to the orientation of the magnetization with respect to the crystallographic axes. As the TAMR effect requires only a single magnetic electrode, in contrast to the tunneling magnetoresistance effect, it offers an attractive route to alternative spintronic applications. In this work we consider the TAMR effect at the single-atom limit by investigating the anisotropy of the local density of states (LDOS) in the vacuum above transition-metal adatoms adsorbed on a noncollinear magnetic surface, the monolayer of Mn on W(1 1 0). This surface presents a cycloidal spin spiral ground state with an angle of 173° between neighboring spins and thus allows a quasi-continuous exploration of the angular dependence of the TAMR of adsorbed adatoms using scanning tunneling microscopy. Using first-principle calculations, we investigate the TAMR of Co, Rh and Ir adatoms on Mn/W(1 1 0) and relate our results to the magnetization-direction-dependent changes in the LDOS. The anisotropic effect is found to be enhanced dramatically on the adsorption of heavy transition-metal atoms, with values of up to 50% predicted from our calculations. This effect will be measurable even with a non-magnetic STM tip.