An observer in uniform linear acceleration responds to the Minkowski vacuum thermally, in the Unruh temperature T_U =\frac{proper\, acceleration}{2 \pi}. An observer in uniform circular motion experiences a similar Unruh-type temperature T_c, with better prospects of detection in analogue spacetime laboratory experiments, but T_c depends not just on the proper acceleration but also on the orbital radius and on the excitation energy. We establish a range of analytic and numerical results for T_c for a massless scalar field in 3+1 and 2+1 spacetime dimensions, the latter being motivated by proposed condensed matter experiments. In particular, we find that the circular motion analogue Unruh temperature grows arbitrarily large in the near-sonic limit, encouragingly for the experimental prospects, but less quickly in effective spacetime dimension 2+1 than in 3+1. [Based on Biermann et al, Phys. Rev. D 102, 085006 (2020)]