This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation (meaning that the system's parameters depend on an autonomously evolving background process). In previous papers on (primarily single-node) queues with Markov modulation, two variants were distinguished: one in which the server speed is modulated, and one in which the service requirement is modulated (i.e., depends on the state of the background process upon arrival). The setup of the present paper, however, is more general, as we allow both the server speed and the service requirement to depend on the background process. For this model we derive a Functional Central Limit Theorem: we show that, after accelerating the arrival processes and the background process, a centered and normalized version of the network population vector converges to a multivariate Ornstein-Uhlenbeck process. The proof of this result relies on expressing the queueing process in terms of Poisson processes with a random time change, an application of the Martingale Central Limit Theorem, and continuous-mapping arguments.