You're talking about temporal compositing. This is a major part of MODIS data because the data is collected so frequently (every 36 hours at most). What's tricky is how to choose one observation to use to represent a week or a month of input data. The MODIS data processing algorithms take into account all sorts of other data in order to choose the highest quality observation. These data include elevation, slope, snow cover, land cover, cloud cover, water vapor, satellite view angle, etc. For the MOD11 data products, these ancillary input data are detailed in section 3.2.6 of the MOD11 Algorithm Theoretical Basis Document. However, the complexity of these compositing algorithms is usually beyond the abilities of anyone except the data product PIs and their grad students.
If your project doesn't require super careful data processing, you can try combining the weekly rasters into one monthly raster yourself. Using Raster Calculator in ArcGIS, Band Math in ENVI, or the
numpy.maximum function in NumPy, you could, on a pixel-by-pixel (i.e. element-wise) basis, take the average, median, mininum, or maximum observation. Of course, this can distory or skew the data depending on your study area, and the element-wise processing can lead to a noisy or patchwork look in the final data.
I hope that helps!