In smart grids, smart meters periodically collect users' fine-grained multi-dimensional energy data, which poses great concerns on users' privacy and security. Existing privacy preserving multi-dimensional aggregation schemes suffer from heavy computational burdens, especially for smart meters with limited computational resources. To address these limitations, in this paper we propose an efficient and flexible multi-dimensional data aggregation scheme called MDA-SMuSha, by which smart meters employ the Shamir's multi-secret sharing to generate a set of shared secrets, with the first one kept locally, while the remained ones are packaged and then uploaded to a control center via an aggregator. By the MDA-SMuSha scheme, aggregation results of smart meters' multi-dimensional energy data during multiple periods can be obtained, with only one time of Paillier encryption conducted on the smart meters. In addition, it allows the control center to send query requests flexibly, i.e., at a pre-specified frequency or whenever it wants to obtain statistical data of interests. Rigorous security analyses show that the MDA-SMuSha scheme satisfies security requirements of privacy-preservation, authenticity and data integrity as well as fault-tolerance. Both theoretical analyses and experiment results show that the MDA-SMuSha scheme outperforms state-of-the art methods in terms of computation costs, with comparable communication costs.