The Terra/Aqua MODerate resolution Imaging Spectroradiometer (MODIS) data have been used widely for global monitoring of the Earth's surface due to their daily fine temporal resolution. The spatial resolution of MODIS time-series (i.e., 500 m), however, is too coarse for local monitoring. A feasible solution to this problem is to downscale the coarse MODIS images, thus creating time-series images with both fine spatial and temporal resolutions. Generally, the downscaling of MODIS images can be achieved by fusing them with fine spatial resolution images (e.g., Landsat images) using spatio-temporal fusion methods. Among the families of spatio-temporal fusion methods, spatial unmixing-based methods have been applied widely owing to their lighter dependence on the available fine spatial resolution images. However, all techniques within this class of method suffer from the same serious problem, that is, the block effect, which reduces the prediction accuracy of spatio-temporal fusion. To our knowledge, almost no solution has been developed to tackle this issue directly. To address this need, this paper proposes a blocks-removed spatial unmixing (SU-BR) method, which removes the blocky artifacts by including a new constraint constructed based on spatial continuity. SU-BR provides a flexible framework suitable for any existing spatial unmixing-based spatio-temporal fusion method. Experimental results on a heterogeneous region, a homogeneous region and a region experiencing land cover changes show that SU-BR removes the blocks effectively and increases the prediction accuracy obviously in all three regions. SU-BR also outperforms two popular spatio-temporal fusion methods. SU-BR, thus, provides a crucial solution to overcome one of the longest standing challenges in spatio-temporal fusion. © 2021 Elsevier Inc.