Inertial measurement units are commonly used to estimate the attitude of moving objects. Numerous nonlinear filter approaches have been proposed for solving the inherent sensor fusion problem. However, when a large range of different dynamic and static rotational and translational motions is considered, the attainable accuracy is limited by the need for situation-dependent adjustment of accelerometer and gyroscope fusion weights. We investigate to what extent these limitations can be overcome by means of artificial neural networks and how much domain-specific optimization of the neural network model is required to outperform the conventional filter solution. A diverse set of motion recordings with a marker-based optical ground truth is used for performance evaluation and comparison. The proposed neural networks are found to outperform the conventional filter across all motions only if domain-specific optimizations are introduced. We conclude that they are a promising tool for inertial-sensor-based real-time attitude estimation, but both expert knowledge and rich datasets are required to achieve top performance.