In this paper, we present a novel approach for a trainable rotation invariant detection of complex structures in 3D microscopic multichannel data using a nonlinear filter approach. The basic idea of our approach is to compute local features in a window around each 3D position and map these features by means of a nonlinear mapping onto new local harmonic descriptors of the local window. These local harmonic descriptors are then combined in a linear way to form the output of the filter. The optimal combination of the computed local harmonic descriptors is determined in previous training step, and allows the filter to be adapted to an arbitrary structure depending on the problem at hand. Our approach is not limited to scalar-valued images and can also be used for vector-valued (multichannel) images such as gradient vector flow fields. We present realizations of a scalar-valued and a vector-valued multichannel filter. Our proposed algorithm was quantitatively evaluated on colorectal cancer cell lines (cells grown under controlled conditions), on which we successfully detected complex 3D mitotic structures. For a qualitative evaluation we tested our algorithms on human 3D tissue samples of colorectal cancer. We compare our results with a steerable filter approach as well as a morphology-based approach.