Advection-diffusion-reaction equations have a moltitudine of applications, such as in climate, water and air quality models, or in short and medium range weather forecasting. Due to the potentially very large number of equations of this kind that have to be solved in order to describe such physical processes, every efficiency gain in the numerical discretization used for this very classical problem is of great practical importance. We propose a fully semi-Lagrangian method for the numerical solution of advection-diffusion-reaction equations that employs a second order semi-Lagrangian scheme. Standard interpolation procedures are used for reconstructiong the solution in the foot of the characteristics, using both structured and unstructured meshes for the space discretization. We also propose a numerical treatment of Dirichlet boundary conditions. The method allows for large time steps, while avoiding the solution of large linear systems, since it follows an explicit approach. The work is completed by numerical experiments that demonstrate the effectiveness of the proposed approach.