Visit one of the three major shrines Kanda Myojin, Hie Jinja or Yasukuni Shrine during the New Year holiday if you happen to be visiting, or immerse yourself in one of the major festivals which takes place during the year. There are no restrictions on entry regardless of religious belief or background.
As stated in the See section, cherry blossoms can be viewed in various places around the Imperial Palace gardens. Also, many people take advantage of the fact that no public roads enter the palace grounds and thus, no red lights for pedestrians for a little jogging around the garden's outer ring which may take roughly an hour to walk around. However, since most of central Chiyoda Ward is office buildings, it is virtually lifeless after sundown as most of the population leaves for the suburbs.
