The architectures of deep neural networks (DNN) rely heavily on the underlying grid structure of variables, for instance, the lattice of pixels in an image. For general high dimensional data with variables not associated with a grid, the multi-layer perceptron and deep belief network are often used. However, it is frequently observed that those networks do not perform competitively and they are not helpful for identifying important variables. In this paper, we propose a framework that imposes on blocks of variables a chain structure obtained by step-wise greedy search so that the DNN architecture can leverage the constructed grid. We call this new neural network Deep Variable-Block Chain (DVC). Because the variable blocks are used for classification in a sequential manner, we further develop the capacity of selecting variables adaptively according to a number of regions trained by a decision tree. Our experiments show that DVC outperforms other generic DNNs and other strong classifiers. Moreover, DVC can achieve high accuracy at much reduced dimensionality and sometimes reveals drastically different sets of relevant variables for different regions.