Content-Centric Networking (CCN) is an alternative to host-centric networking exemplified by today's Internet. CCN emphasizes content distribution by making content directly addressable. Named-Data Networking (NDN) is an example of CCN being considered as a candidate next-generation Internet architecture. One key NDN feature is router-side content caching that optimizes bandwidth consumption, reduces congestion and provides fast fetching for popular content. Unfortunately, the same feature is also detrimental to privacy of both consumers and producers of content. As we show in this paper, simple and difficult-to-detect timing attacks can exploit NDN routers as "oracles" and allow the adversary to learn whether a nearby consumer recently requested certain content. Similarly, probing attacks that target adjacent content producers can be used to discover whether certain content has been recently fetched. After analyzing the scope and feasibility of such attacks, we propose and evaluate some efficient countermeasures that offer quantifiable privacy guarantees while retaining key features of NDN.