Recently, a new relaying strategy, quantize-map-and-forward (QMF) scheme, has been demonstrated to approximately achieve (within an additive constant number of bits) the Gaussian relay network capacity, universally, i.e., for arbitrary topologies, channel gains, and SNRs. This was established using Gaussian codebooks for transmission and random mappings at the relays. In this paper, we develop ... [Show full abstract] structured lattice codes that implement the QMF strategy. The main result of this paper is that such structured lattice codes can approximately achieve the Gaussian relay network capacity universally, again within an additive constant. In addition, we establish a similar result for half-duplex networks, where we demonstrate that one can approximately achieve the capacity using fixed transmit-receive (TX-RX) schedules for the relays with no transmit power optimization across the different TX-RX states of the network.