Echoing what Pyron is saying, most of the lines that run through central Tokyo are Tokyo Metro rather than JR. As Pyron can tell you, my home is nowhere near a JR station, and I live about as centrally as you can get. (I have a view of Tokyo Tower from my living room.) As a resident of Tokyo, I think I ride the JR at most once or twice a month and that's usually to go to places like Costco.
However, many great tourist spots are accessible by JR. If you're traveling outside of Tokyo frequently, then I think the JR pass could be worth your money. Otherwise, I'd just stick to the Suica/Passmo (the cards that you touch down on the reader at the gates) and refill those as you go.
Edit: I forgot that you could walk to Tokyo Tower from Hamamatsucho which is a JR station.