Most wind turbines are remotely monitored 24/7 to allow for early detection of operation problems and developing damage. We present a new fault detection approach for vibration-monitored drivetrains that does not require any feature engineering. Our method relies on a simple model architecture to enable a straightforward implementation in practice. We propose to apply convolutional autoencoders for identifying and extracting the most relevant features from a broad continuous range of the spectrum in an automated manner, saving time and effort. We focus on the range of [0, 1000] Hz for demonstration purposes. A spectral model of the normal vibration response is learnt for the monitored component from past measurements. We demonstrate that the trained model can successfully distinguish damaged from healthy components and detect a damaged generator bearing and damaged gearbox parts from their vibration responses. Using measurements from commercial wind turbines and a test rig, we show that vibration-based fault detection in wind turbine drivetrains can be performed without the usual upfront definition of spectral features. Another advantage of the presented method is that a broad continuous range of the spectrum can be monitored instead of the usual focus on monitoring individual frequencies and harmonics. Future research should investigate the proposed method on more comprehensive datasets and fault types.