We propose an approach to upscale solute transport in spatially periodic porous media. Our methodology relies on pore scale information to predict large scale transport features, including explicit reconstruction of the solute plume, breakthrough curves at fixed distances, and spatial spreading transverse to the main flow direction. The proposed approach is grounded on the recently proposed trajectory-based Spatial Markov model (tSMM), which upscales transport based on information collected from advective-diffusive particle trajectories across one periodic element. In previous works, this model has been applied solely to one-dimensional transport in a single periodic pore geometry. In this work we extend the tSMM to the prediction of multi-dimensional solute plumes. This is obtained by analyzing the joint space-time probability distribution associated with discrete particles, as yielded by the tSMM. By comparing numerical results from fully resolved simulations and predictions obtained with the tSMM over a wide range of Péclet numbers, we demonstrate that the proposed approach is suitable for modeling transport of conservative and linearly decaying solute species in a realistic pore space and showcase the applicability of the model to predict steady state solute plumes. Additionally, we evaluate the model performance as a function of numerical parameters employed in the tSMM parameterization.