What is the best solution to drying hair after a FUE hair transplant?

It is best to contact your surgeon for post surgery issues and care since all doctors have slightly different instructions.You can lightly pat dry.

I would recommend calling your hair restoration clinic and following their post op instructions for drying your scalp without effecting the grafts. 

You can now 2 weeks post op towel dry your hair, but I still wouldn't recommend scrubbing until all of the scabs have fallen out.  When you wash your hair you can use the balls of your fingertips to lightly massage the scabs to loosen them up.  Some of the scabs could stay in place for up to a month so be careful not to force them off.  

You can wash your hair as per usual by week 2. The grafts have settled and you can wash your hair using shampoo and towel dry. All of the scabs should be washed off at this point as well. 

Place the shampoo on your head and let it sit for 10 minutes so that the crusts get water logged, then gently with your finger tips, rub them off. Do not be aggressive, but repeat this daily until they are all off.  A towel gently pressed to dry your hair and even a light blow not hot dryer will work