Primary progressive aphasia is a clinical syndrome defined by progressive deficits isolated to speech and/or language, and can be classified into non-fluent, semantic and logopenic variants based on motor speech, linguistic and cognitive features. The connected speech of patients with primary progressive aphasia has often been dichotomized simply as ‘fluent’ or ‘non-fluent’, however fluency is a multidimensional construct that encompasses features such as speech rate, phrase length, articulatory agility and syntactic structure, which are not always impacted in parallel. In this study, our first objective was to improve the characterization of connected speech production in each variant of primary progressive aphasia, by quantifying speech output along a number of motor speech and linguistic dimensions simultaneously. Secondly, we aimed to determine the neuroanatomical correlates of changes along these different dimensions. We recorded, transcribed and analysed speech samples for 50 patients with primary progressive aphasia, along with neurodegenerative and normal control groups. Patients were scanned with magnetic resonance imaging, and voxel-based morphometry was used to identify regions where atrophy correlated significantly with motor speech and linguistic features. Speech samples in patients with the non-fluent variant were characterized by slow rate, distortions, syntactic errors and reduced complexity. In contrast, patients with the semantic variant exhibited normal rate and very few speech or syntactic errors, but showed increased proportions of closed class words, pronouns and verbs, and higher frequency nouns, reflecting lexical retrieval deficits. In patients with the logopenic variant, speech rate (a common proxy for fluency) was intermediate between the other two variants, but distortions and syntactic errors were less common than in the non-fluent variant, while lexical access was less impaired than in the semantic variant. Reduced speech rate was linked with atrophy to a wide range of both anterior and posterior language regions, but specific deficits had more circumscribed anatomical correlates. Frontal regions were associated with motor speech and syntactic processes, anterior and inferior temporal regions with lexical retrieval, and posterior temporal regions with phonological errors and several other types of disruptions to fluency. These findings demonstrate that a multidimensional quantification of connected speech production is necessary to characterize the differences between the speech patterns of each primary progressive aphasic variant adequately, and to reveal associations between particular aspects of connected speech and specific components of the neural network for speech production.