Improving energy efficiency of Heating, Ventilation and Air Conditioning (HVAC) systems is a primary objective for the society. Model Predictive Control (MPC) techniques for HVAC systems have recently received particular attention, since they can naturally account for several factors, such as weather and occupancy forecasts, comfort ranges and actuation constraints. Developing effective MPC based control strategies for HVAC systems is nontrivial, since buildings dynamics are nonlinear and affected by various uncertainties. Further, the complexity of the MPC problem and the burden of on-line computations can lead to difficulties in integrating this scheme into a building management system. We propose to address this computational issue by designing a scenario-based explicit MPC strategy, i.e., a controller that is simultaneously based on explicit representations of the MPC feedback law and accounts for uncertainties in the occupancy patterns and weather conditions by using the scenarios paradigm. The main advantages of this approach are the absence of a-priori assumptions on the distributions of the uncertain variables, the applicability to any type of building, and the limited on-line computational burden, enabling practical implementations on low-cost hardware platforms. We illustrate the practical implementation of the proposed explicit MPC controller on a room of a university building, showing its effectiveness and computational tractability.