If I understand your question correctly I have done something similar before; just had to remember what I did.
This will work if your polys are overlapping AND not the same layer. If they are on the same layer and it it is possible, move them to seperate layers, if you don't the target layer will just inherit one of the values.
The key process to use is the Join attributes by location under the Vector > Data Mangement menu. Here specify your hexagonal field as the target layer and the join layer as one of your polygons. Name your resulting layer and tick the radio button for Keep All Records.
The resulting layer should be a copy of your hexgonal layer with all the cells that intersected your polygon layer. Now for the interation use your last resulting layer as the target layer and join the next polygon layer. Take the resulting lasyer and join your next layer until you have join all your layers.
Your final layer should have inherited all the attribute fields of the previous layers and much of it may say NULL.
This is where we have to play in some of the guts of a shapefile, so if you havn't saved the layer as a shapefile do so now. And remove the layer from your workspace in QGIS.
Open the shapefile's *.dbf files in OpenOffice Calc, DO NOT use Excel. It will mangle the file and it useless, learned that the hard way. And do not reorder the data, it is very specific on the order of rows in the file as well. Check to make sure the NULLs now are Zeros if they are not, replace them carefully.
Now insert the shapefile into QGIS and open the Attribute Table. It should now show zeros instead of NULL. Use the Field Calculator and sum the fields that are realted to your work in a new column.
Now you have a summed field that should show with a degree of accuracy the values of the all the feature collections overlaping the target field. You can use the Symbology function to color your polygons based upon the value of your summed field for a nice choropleth map.
If you want to rid of all the unused field in the file. Just open the dbf file again in Calc and delete the unwanted columns at will. Personally, I'd advise learning to get comfortable going into a shapefile this way as I do it often.