In this paper, the consensus problem for high-order discrete-time networked multi-agent systems (D-NMAS) is investigated by distributed feedback protocols. By constructing the self-feedback matrix and the neighbouring feedback matrix for networks, consensus protocols are designed under three different cases and the corresponding convergence analysis is provided. Consensus convergence results of networks are provided by three final consensus values, which are related to self-feedback matrices, initial states of networks and the topology of networks, not related to time delays. In the first case where a directed network with a fixed topology is concerned, the high-order discrete-time consensus problem is studied as an example, and a sufficient and necessary condition is obtained. In the scenario with directed networks with switching topology, a sufficient condition guaranteeing the consensus of high-order D-NMAS is derived, after the consensus analysis is transformed into stability analysis. As for directed networks with switching topology and time delays, the discrete-time stability model with time delays is converted into a general discrete-time stability model by an augmented method and the sufficient condition is provided to achieve consensus for directed networks. Furthermore, the sufficient conditions determining the neighbouring feedback matrix are independent of the number of agents. Two numerical examples are provided to demonstrate the correctness and effectiveness of the theoretical results.