Even though Israel controls the Old Jerusalem and the Temple Mount, the mount itself is actually administered by the Muslims. Many Orthodox Jews will not go up on the mount for fear of desecrating the spot on which the temple once stood as no one knows the exact location of the ancient temple.
Here is a quote from Wikipedia,
Controlled by Israel since 1967, both Israel and the Palestinian Authority claim sovereignty over the site, which remains a key issue in the Arab-Israeli conflict. A Muslim council, known as the Muslim Waqf, manages the site. The Israeli government enforces a controversial ban on prayer by non-Muslim visitors.
No one except Muslims are allowed to pray or worship on the Mount. If anyone tries to pray, sing, or worship God in any manner, they will be expelled from the Mount by Muslim guards that patrol the that area of approximately 37 acres that comprises the Temple Mount.
For those that are interested there is a 90 minute video of Rabbi Richmond, from the Temple Institute, who toured the Temple Mount with a video camera. Click here for the introduction and to view the video.