EDIT: The Dengue Fever scare of 2014 is now over!
Normally I would recommend checking out the nearby Shinjuku Chuo Park (aka Shinjuku Central Park) that borders both the Hyatt Regency Tokyo and the Park Hyatt Tokyo during the course of your stay. However, there are now reports that someone may have contracted dengue fever via a bite from a mosquito in this park.
Dengue fever, which has not had a major presence in Japan since the end of World War II, is a particularly nasty illness where symptoms can range from “fever, headache, muscle and joint pains, and a characteristic skin rash that is similar to measles.” In a small percentage of people, it can lead to far more serious symptoms, possibly even death.
Events at the park have already been cancelled. You should have nothing to worry about in the general vicinity of either hotel, but I would highly recommend avoiding going into the park until at least late fall, when the cold weather has settled in. If you must pass through for some reason, cover up and use bug repellent over any exposed skin. Better safe than sorry!