Looks to me like your standard deviations are being calculated from a single pixel (the center of each large cell). This will always give you zero - no variation from one value! It worked for you with polygons because by default the cell size used was your value grid cell size.
The default cell size used is usually the max of the inputs (this varies by tool). If you set the geoprocessing environment cellsize to the value grid before the tool is run, you may get the results you're looking for.
Of course this means your std value is replicated for all cells in the zone - this seems very strange until you figure out that this allows zonal statistics to be used as part of a complex map algebra expression, for example, to identify all the cells in each zone that share the maximum zone value:
ZonalStatistics(zonegrid, "VALUE", valuegrid, "MAXIMUM") == valuegrid
I would be remiss to mention that statistics calculated using the zonal tools are by definition area (cell area) weighted - it is best practice to get a more accurate weighting by projecting the raster data before doing these calculations.
