If shaking hands feels good to you, go for it! People in the U.S. tend to get a little lazy about it, but for most people, it's good to have more opportunities for touch with other human beings.
(The only exception would be someone suffering from OCD who has a contamination phobia. But you will notice that pretty quickly if you try holding out your hand to the person.)
Example: a few weeks ago my son and I felt fortunate when a medical specialist squeezed us into her already busy schedule. I shook her hand at the end of the appointment and thanked her by name, and I encouraged my son to do the same, since we both felt that it had been one of the most helpful and pleasant medical appointments we had ever experienced. She seemed a bit surprised, but pleased. My son and I felt good about it, because we really wanted to show her how grateful we were.
When we went for the follow-up appointment two weeks later, this doctor herself took the initiative to shake our hands at the end of the appointment. (This time we were a bit surprised -- but also pleased.) I think this meant that the contact felt good to her.
So -- if you like shaking hands, don't let social hierarchy or cultural customs hold you back.