We propose a domino logic architecture for memristor-based neuromorphic computing. The design uses the delay of memristor RC circuits to represent synaptic computations and a simple binary neuron activation function. Synchronization schemes are proposed for communicating information between neural network layers, and a simple linear power model is developed to estimate the design's energy efficiency for a particular network size. Results indicate that the proposed architecture can achieve 0.61 fJ per classification per component (neurons and synapses) and outperforms other designs in terms of energy per % accuracy.