This paper proposes a distributed hierarchical cooperative (DHC) control strategy for a cluster of islanded microgrids (MGs) with intermittent communciation, which can regulate the frequency/voltage of all distributed generators (DGs) within each MG as well as ensure the active/reactive power sharing among MGs. A droop-based distributed secondary control (DSC) scheme and a distributed tertiary control (DTC) scheme are presented based on the iterative learning mechanics, by which the control inputs are merely updated at the end of each round of iteration, and thus each DG only needs to share information with its neighbors intermittently in a low-bandwidth communication manner. A two-layer sparse communication network is modeled by pinning one or some DGs (pinned DGs) from the lower network of each MG to constitute an upper network. Under this control framework, the tertiary level generates the frequency/voltage references based on the active/reactive power mismatch among MGs while the pinned DGs propagate these references to their neighbors in the secondary level, and the frequency/voltage nominal set-points for each DG in the primary level can be finally adjusted based on the frequency/voltage errors. Stability analysis of the two-layer control system is given, and sufficient conditions on the upper bound of the sampling period ratio of the tertiary layer to the secondary layer are also derived. The proposed controllers are distributed, and thus allow different numbers of heterogeneous DGs in each MG. The effectiveness of the proposed control methodology is verified by the simulation of an AC MG cluster in Simulink/SimPowerSystems.