K.H.
For whiter whites WITHOUT bleach try
1) vinegar and hot water setting
2) oxyclean and hot water setting (I'd read the directions carefully with that one if the sheets are expensive)
3) Baking soda
4) The sun, you'll probably have to wait until summer for this one but the sun will "bleach" whites nicely without need for any chemicals at all. Try the other 3 for now though, til summer comes around again:)
K. H