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This study is concerned with cascade architectures of fuzzy neural networks. These networks exhibit three interesting and practically appealing features: (i) come with sound and transparent logic characteristics by being developed with the aid of AND and OR fuzzy neurons and subsequently logic processors (LPs), (ii) possess significant learning abilities and in this way fall in the realm of neuro-fuzzy architectures, and (iii) exhibit an evident hierarchical structure owing to the cascade of the LPs. We discuss main functional properties of the model and relate them to its form of cascade-type of systems formed as a stack of LPs. The construction of the systems of this form calls for some structural optimization that is realized in the realm of genetic optimization. The structure of the network that deals with a selection of a subset of input variables and their distribution across the individual LPs is optimized with the use of genetic algorithms (GAs). The chromosomes encode the order of the variables as well as include the parameters (connections) of the neurons. We discuss various schemes of genetic optimization (both a two-level and single-level GA) and gradient-based learning aimed at further refinement of the connections of the neurons. We elaborate on the interpretation aspects of the network and show how this leads to a Boolean or multivalued logic description of the experimental data. A number of numeric data sets are discussed with respect to the performance of the constructed networks and their interpretability.