My three year old daughter has had a persistent dry cough and minimal runny nose without a fever.

Possible allergies.: If your daughter is having recurrent symptoms with out fevers and signs of acute infection she is most likely having recurrent allergic symptoms. I would rec using saline nose drops for nasal hygiene. Take some loratidine otc, and look for signs of improvement. If nothing changes in 3-4 weeks then talk to your pediatrician.