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We consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multipliers (ADMM) with three blocks converges globally to their KKT points under some technical conditions. In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan’s assumptions. Moreover, in order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions.