In this paper, we propose a novel randomized approach to Stochastic Model Predictive Control (SMPC) for a linear system affected by a disturbance with unbounded support. As it is common in this setup, we focus on the case where the input/state of the system are subject to probabilistic constraints, i.e., the constraints have to be satisfied for all the disturbance realizations but for a set having probability smaller than a given threshold. This leads to solving at each time t a finite-horizon chance-constrained optimization problem, which is known to be computationally intractable except for few special cases. The key distinguishing feature of our approach is that the solution to this finite-horizon chance-constrained problem is computed by first extracting at random a finite number of disturbance realizations, and then replacing the probabilistic constraints with hard constraints associated with the extracted disturbance realizations only. Despite the apparent naivety of the approach, we show that, if the control policy is suitably parameterized and the number of disturbance realizations is appropriately chosen, then, the obtained solution is guaranteed to satisfy the original probabilistic constraints. Interestingly, the approach does not require any restrictive assumption on the disturbance distribution and has a wide realm of applicability.