In this paper, we study a class of neural networks with variable coefficients which includes delayed Hopfield neural networks, bidirectional associative memory networks and cellular neural networks as its special cases. By matrix theory and inequality analysis, we not only obtain some new sufficient conditions ensuring the existence, uniqueness, global attractivity and global exponential stability of the periodic solution but also estimate the exponentially convergent rate. Our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Moreover, these conclusions are presented in terms of system parameters and can be easily verified for the globally Lipschitz and the spectral radius being less than 1. Therefore, our results have an important leading significance in the design and applications of periodic oscillatory neural circuits for neural networks with delays.