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Due to recurrent (e.g. daily or weekly) patterns and non-recurrent disruptions (e.g. caused by incidents), travel times in road networks are time-dependent and inherently random. This is challenging for travelers planning a future trip, aiming to ensure on-time arrival at the destination, while also trying to limit the total travel-time budget spent. The focus of this paper lies on determining their optimal departure time: the latest time of departure for which a chosen on-time arrival probability can be guaranteed. To model the uncertainties in the network, a Markovian background process is used, tracking events affecting the driveable vehicle speeds on the links, thus enabling us to incorporate both recurrent and non-recurrent effects. It allows the evaluation of the travel-time distribution, given the state of this process at departure, on each single link. Then, a computationally efficient algorithm is devised that uses these individual link travel-time distributions to obtain the optimal departure time for a given path or origin-destination pair. Since the conditions in the road network, and thus the state of the background process, may change between the time of request and the advised time of departure, we consider an online version of this procedure as well, in which the traveler receives departure time updates while still at the origin. Finally, numerical experiments are conducted to exemplify a selection of properties of the optimal departure time and, moreover, quantify the performance of the presented algorithms in an existing road network -- the Dutch highway network.