The current paper deals with the numerical prediction of the mechanical response of asphalt concretes for road pavements, using artificial neural networks (ANNs). The asphalt concrete mixes considered in this study have been prepared with a diabase aggregate skeleton and two different types of bitumen, namely, a conventional bituminous binder and a polymer-modified one. The asphalt concretes were produced both in a road materials laboratory and in an asphalt concrete production plant. The mechanical behaviour of the mixes was investigated in terms of Marshall stability, flow, quotient, and moreover by the stiffness modulus. The artificial neural networks used for the numerical analysis of the experimental data, of the feedforward type, were characterized by one hidden layer and 10 artificial neurons. The results have been extremely satisfactory, with coefficients of correlation in the testing phase within the range 0.98798–0.91024, depending on the considered model, thus demonstrating the feasibility to apply ANN modelization to predict the mechanical and performance response of the asphalt concretes investigated. Furthermore, a closed-form equation has been provided for each of the four ANN models developed, assuming as input parameters the production process, the bitumen type and content, the filler/bitumen ratio, and the volumetric properties of the mixes. Such equations allow any other researcher to predict the mechanical parameter of interest, within the framework of the present study.