The social media revolution has produced a plethora of web services to which users can easily upload and share multimedia documents. Despite the popularity and convenience of such services, the sharing of such inherently personal data, including speech data, raises obvious security and privacy concerns. In particular, a user's speech data may be acquired and used with speech synthesis systems to produce high-quality speech utterances which reflect the same user's speaker identity. These utterances may then be used to attack speaker verification systems. One solution to mitigate these concerns involves the concealing of speaker identities before the sharing of speech data. For this purpose, we present a new approach to speaker anonymization. The idea is to extract linguistic and speaker identity features from an utterance and then to use these with neural acoustic and waveform models to synthesize anonymized speech. The original speaker identity, in the form of timbre, is suppressed and replaced with that of an anonymous pseudo identity. The approach exploits state-of-the-art x-vector speaker representations. These are used to derive anonymized pseudo speaker identities through the combination of multiple, random speaker x-vectors. Experimental results show that the proposed approach is effective in concealing speaker identities. It increases the equal error rate of a speaker verification system while maintaining high quality, anonymized speech.