A new deep neural network (NN) approach applied to antenna array adaptive beamforming is presented in this paper. A recurrent NN (RNN) based on the gated recurrent unit (GRU) architecture is used as a beamformer in order to produce proper complex weights for the feeding of the antenna array. The proposed RNN utilizes four hidden GRU layers and one extra layer for linear transformation. The produced weights are subsequently compared with respective weights derived by a null steering beamforming (NSB) technique in order to measure the accuracy of the RNN. The RNN training is performed by using a large data set derived from an NSB technique applied to a realistic microstrip linear antenna array, in order to take into account real-world effects, like the non-isotropic radiation pattern of an array element and the mutual coupling between the array elements. The RNN performance is examined by using the root mean square error metric, whereas its beamforming performance is evaluated by estimating the mean value and the standard deviation of the divergences of the main lobe and nulls directions from their respective desired directions. A comparison between various NN structures and an overall study of the proposed RNN-based beamformer are also presented.