Remote Data Checking (RDC) is a technique by which clients can establish that data outsourced at untrusted servers remains intact over time. RDC is useful as a prevention tool, allowing clients to periodically check if data has been damaged, and as a repair tool whenever damage has been detected. Initially proposed in the context of a single server, RDC was later extended to verify data integrity in distributed storage systems that rely on replication and on erasure coding to store data redundantly at multiple servers. Recently, a technique was proposed to add redundancy based on network coding, which offers interesting tradeoffs because of its remarkably low communication overhead to repair corrupt servers. Unlike previous work on RDC which focused on minimizing the costs of the prevention phase, we take a holistic look and initiate the investigation of RDC schemes for distributed systems that rely on network coding to minimize the combined costs of both the prevention and repair phases. We propose RDC-NC, a novel secure and efficient RDC scheme for network coding-based distributed storage systems. RDC-NC mitigates new attacks that stem from the underlying principle of network coding. The scheme is able to preserve in an adversarial setting the minimal communication overhead of the repair component achieved by network coding in a benign setting. We implement our scheme and experimentally show that it is computationally inexpensive for both clients and servers.