Circulant and skew-circulant splitting iteration method is employed for solving linear systems from finite difference discretisation of tempered fractional diffusion equations. The method is shown to be convergent unconditionally and the convergence rate is fast in numerical tests. In each iteration, a circulant system and a skew-circulant system are required to be solved which cost only O(Nlog N) operations by fast Fourier transform, where N is the number of interior mesh points in space. Moreover, the induced preconditioner possesses circulant-times-skew-circulant structure so that it can be inverted in O(Nlog N) operation. In all of our numerical experiments, the preconditioner performs very well with a very simple choice of the method parameter.