In this paper, a new data-driven distributed control structure for frequency/voltage regulation and active/reactive power sharing of islanded microgrids is proposed. By using a droop-free concept, the proposed method avoids the inherent timescale separation between primary and secondary control, and improves the transient response for microgrids with inductive, resistive or mixed lines. Besides, the proposed control algorithm is fully data-driven and does not rely on the accurate model or physical parameters of the microgrid. Instead, a classical neural network is adopted to learn the unknown system dynamics online, and an adaptive controller is designed based on the learning results. Because no system model is needed, and the control algorithm can be adjusted in real-time, a remarkable plug-and-play capability is also achieved. The effectiveness of the proposed method is demonstrated via simulations.