3 months after, tranplanted hair still looks the same as day one

It can take 6 to 12 months after surgery to see results.  If you are concerned it is best to contact your doctor.

You are probably not washing the scalp thoroughly. Give the recipient area a good scrub in the shower under water and shampoo, using your finger tips and not your finger nails, so the transplanted hairs shed. 

This is a good question. These hair follicles if they have not shed and are not growing are actually stopping the new hairs from growing in. Normally they should have shed with normal scrubbing of the scalp. We have seen this rare case before and we simply tugged on each transplanted hair follicle and if it pulled out with ease, then the transplanted hair will begin to grow. Sometimes the trapped hairs can cause an inflammatory response and become irritated. You need to see your hair restoration professional. 

It just takes forever for grafts to grow it seems.  Typically the grafts take about 3 months to start producing new hair.  And on top of that, some of the native (non-transplanted hairs) in the recipient area shed due to the manipulation of the scalp and they too take 3-6 months to regrow as.  In short the transplanted area will look worse after 3 months than when you started.  Take heart though.  You should begin to see improvement very soon.  In my experience, 3 months is the low point for density.

I always tell my patients that it is not unusual for you to look the same 3 months after the surgery as before the surgery because it takes at least 4 months for the hair to start growing