I can not get 1km MODIS data aligned with ESRI basemaps in ArcGIS 10.2.2 for Desktop. See the photo below.
I reprojected MODIS17A3 data using the MRT (MODIS reprojection tool). I selected WGS84 under Projection Parameters. I tiled two files (shouldn't matter). My shapefile is in GCS_WGS_84 and it aligns perfectly over the islands. It seems that the MODIS data should move south and east. Generally, it is aligned, but there are some grid cells completely in the water.
I have several study areas in the Great Lakes region. Another is pictured at the bottom. Any ideas? These are quite north of the equator.