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This paper addresses the design and performance evaluation of relay strategies for opportunistic Delay Tolerant Networks (DTNs) augmented with throwboxes. By opportunistic we mean that a node does not have any knowledge regarding its past and future contact opportunities with the other nodes. We consider a network model composed of both mobile relay nodes and throwboxes, where throwboxes are stationary wireless devices acting simply as fixed relays. We propose and evaluate various relay strategies, where the goal is to take advantage of the presence of throwboxes to minimize resources consumption at mobile nodes. Under Markovian assumptions we introduce a mathematical framework which allows us to calculate the main performance metrics (average delivery delay, overhead, etc.) of each proposed relay scheme. The obtained results highlight the various trade-offs that are left to network designers when adding throwboxes to a DTN, and draw insights on the effectiveness of these strategies.