If you're staying at some luxury hotels, there is an alternative. At the Peninsula, your room phone will work as a mobile throughout the city. At the Park Hyatt, although it's not publicized, if you ask, they will loan you a phone.
In both cases, I believe there is no rental fee and you only pay for outbound calls at their rate plus 20%, which works out to about Y80 a minute. Beats the heck out of renting a SIM card.
Another way of course is to get a Japanese friend to buy a prepaid SIM for you. You can recharge it anywhere without any id requirement.
The regulation prohibiting visitors from buying prepaid SIMs is as idiotic in origin as the above posters say.