When working with continuous rasters, such as rainfall or elevation, try the "Cubic" Resampling Technique. This calculates the value of each pixel by fitting a smooth curve based on the surrounding 16 pixels. It produces the smoothest image but can create values outside of the range found in the source data - and in this way is best suited to continuous rasters as opposed to discrete rasters such as those based on classed attributes, eg. land cover, where surrounding pixel values may be totally independent from each other.
This method was successful for me when reducing the cell size of a rainfall raster from 2.5km to 1km so it would align with a 1km polygon grid and therefore process my zonal statistics (as table) completely. Without doing this, only 40% of the zones were processed.