In this work, a cost-effective computational framework is proposed to investigate the effect of random microstructural defects on the elastic properties of materials additively manufactured using the Fused Deposition Modeling (FDM) technique. Random fields of the apparent elasticity tensor of the 3D printed material are computed using the moving window technique and stochastic homogenization based on the extended finite element method (XFEM) coupled with Monte Carlo simulation. The implementation of XFEM is particularly suitable for this type of problem since there is no need to generate FE meshes conforming to internal boundaries (material interfaces) in the model, thus leading to savings of computing time. The random spatial variation of the elastic properties of the FDM printed material is caused by the existence of random microstructural defects. These are due to morphological irregularities in the shape of the pores, which are formed by deviations in the spatial placement of the filament lines. The moving window technique is used to extract statistical volume elements (SVEs) of the mesoscale structure of the 3D printed material. XFEM analysis of the SVE models is subsequently performed to obtain the upper and lower bounds of their elastic properties by solving multiple boundary value problems under kinematic and static uniform boundary conditions. The effect of scale factor, inter- and intra-layer overlap of printed filament lines on the mesoscale random fields is assessed and useful conclusions are derived regarding the degree of variability and anisotropy in different cases.