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Multi-output Gaussian process (GP) regression has been widely used as a flexible nonparametric Bayesian model for predicting multiple correlated outputs given inputs. However, the cubic complexity in the sample size and the output dimensions for inverting the kernel matrix has limited their use in the large-data regime. In this paper, we introduce the factorial stochastic differential equation as a representation of multi-output GP regression, which is a factored state-space representation as in factorial hidden Markov models. We propose a structured mean-field variational inference approach that achieves a time complexity linear in the number of samples, along with its sparse variational inference counterpart with complexity linear in the number of inducing points. On simulated and real-world data, we show that our approach significantly improves upon the scalability of previous methods, while achieving competitive prediction accuracy.