Microstructure reconstruction is an important cornerstone to the inverse materials design concept. In this work, a general algorithm is developed to reconstruct a three-dimensional microstructure from given descriptors. Based on two-dimensional (2D) micrographs, this reconstruction algorithm allows valuable insight through spatial visualization of the microstructure and in silico studies of structure-property linkages. The formulation ensures computational efficiency by casting microstructure reconstruction as a gradient-based optimization problem. Herein, the descriptors can be chosen freely, such as spatial correlations or Gram matrices, as long as they are differentiable with respect to the microstructure. Because real microstructure samples are commonly available as 2D microscopy images only, the desired descriptors for the reconstruction process are prescribed on orthogonal 2D slices. This adds a source of noise, which is handled in a new, superior and interpretable manner. The efficiency and applicability of this formulation is demonstrated by various numerical experiments.