Automated speech analysis is a thriving approach to detect early markers of Alzheimer's disease (AD). Yet, recording conditions in most AD datasets are heterogeneous, with patients and controls often evaluated in different acoustic settings. While this is not a problem for analyses based on speech transcription or features obtained from manual alignment, it does cast serious doubts on the validity of acoustic features, which are strongly influenced by acquisition conditions. We examined this issue in the ADreSSo dataset, derived from the widely used Pitt corpus. We show that systems based on two acoustic features, MFCCs and Wav2vec 2.0 embeddings, can discriminate AD patients from controls with above-chance performance when using only the non-speech part of the audio signals. We replicated this finding in a separate dataset of Spanish speakers. Thus, in these datasets, the class can be partly predicted by recording conditions. Our results are a warning against the use of acoustic systems for identifying patients based on non-standardized recordings. We propose that acoustically heterogeneous datasets for dementia studies should be either (a) analyzed using only transcripts or other features derived from manual annotations, or (b) replaced by datasets collected with strictly controlled acoustic conditions.