In recent years there has been growing popularity of leveraging cloud computing for storing and querying attributed graphs, which have been widely used to model complex structured data in various applications. Such trend of outsourced graph analytics, however, is accompanied with critical privacy concerns regarding the information-rich and proprietary attributed graph data. In light of this, we design, implement, and evaluate OblivGM, a new system aimed at oblivious graph analytics services outsourced to the cloud. OblivGM focuses on the support for attributed subgraph matching, one popular and fundamental graph query functionality aiming to retrieve from a large attributed graph subgraphs isomorphic to a small query graph. Built from a delicate synergy of insights from attributed graph modelling and advanced lightweight cryptography, OblivGM protects the confidentiality of data content associated with attributed graphs and queries, conceals the connections among vertices in attributed graphs, and hides search access patterns. Meanwhile, OblivGM flexibly supports oblivious evaluation of varying subgraph queries, which may contain equality and/or range predicates. Extensive experiments over a real-world attributed graph dataset demonstrate that while providing strong security guarantees, OblivGM achieves practically affordable performance (with query latency on the order of a few seconds).