Inertialess anisotropic particles in a Rayleigh-Benard turbulent flow show maximal tumbling rates for weakly oblate shapes, in contrast with the universal behaviour observed in developed turbulence where the mean tumbling rate monotonically decreases with the particle aspect ratio. This is due to the concurrent effect of turbulent fluctuations and of a mean shear flow whose intensity, we show, is determined by the kinetic boundary layers. In Rayleigh-Benard turbulence prolate particles align preferentially with the fluid velocity, while oblate ones orient with the temperature gradient. This analysis elucidates the link between particle angular dynamics and small-scale properties of convective turbulence and has implications for the wider class of sheared turbulent flows.