We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by associ- ating the rotation group with the three-torus through a periodic extension. Our sampling theorem requires 4L3 samples to capture all of the information content of a signal band-limited at L, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as O(L4), compared to the naive scaling of O(L6). For the common case of a low directional band-limit N, complexity is reduced to O(NL3). Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.