This paper investigates the problem of how to efficiently operate Gaussian half-duplex diamond networks with N relays. It derives sufficient conditions that ensure that the network is operated close to its Shannon capacity, with a linear number in N (instead of exponential) of receive/transmit configuration states. Particularly, these states consist of having either at most one relay receiving or at most one relay transmitting. A transmission scheme is also designed and it is shown that, when the aforementioned conditions are satisfied, it achieves a rate that is to within a constant gap of the Shannon capacity. An appealing feature of the proposed scheme is that it offers guidelines on how to route the information through the relays so that the network operates close to its Shannon capacity.