This paper analyzes the iteration-complexity of a class of linearized proximal multiblock alternating direction method of multipliers (ADMM) for solving linearly constrained nonconvex optimization problems. The subproblems of the linearized ADMM are obtained by partially or fully linearizing the augmented Lagrangian with respect to the corresponding minimizing block variable. The derived complexity bounds do not depend on the specific forms of the actual linearizations but only on some Lipschitz constants which quantify the approximation errors. Iteration-complexity is then established by showing that the linearized ADMM class is a subclass of a general non-Euclidean ADMM for which a general iteration-complexity analysis is also obtained. Both ADMM classes allow the choice of a relaxation parameter in the interval (0, 2) as opposed to being equal to one as in many of the previous papers on this topic.