When you're a foreigner in Tokyo you can hang out in one of two types of neighborhoods: areas where lots of other foreigners hang out, and areas where they don't.
If your objective in visiting Tokyo is to pretend that you are, in fact, back in the United States, you'll probably find yourself spending a lot of time in Roppongi. This district grew up around a nearby U.S. military base, now defunct, so Roppongi resembles Sunset Strip more than the Silk Road.
Replete with the full run of American eateries, from Johnny Rockets to Spago, Roppongi is the drinking and discoing haven where English teachers and expat accountants alike boogie the night away, free to besod themselves far past the point of social reprobation at home.
Also on the foreigner-friendly front, though more from a daytime-shopping-and-strolling-as-opposed-to-nocturnal- boozing and-barfing angle, Harajuku and Omotesando form a sort of Melrose Avenue (to continue the L.A. analogy) of Tokyo, replete with sidewalk cafes and designer boutiques. Harajuku is a happy place on weekends especially, when numerous bands perform for free in Yoyogi Park.
More Japanese in spirit are Shinjuku and Shibuya -- the mid-town Manhattan and Greenwich Village of Tokyo, respectively, but a lot more crowded. Also, a lot more futuristic. With massive neon signs and three-story-high TV screens glowing through the haze, either ward could have served as the set of Blade Runner. Shibuya is more of a young hipster's scene and features perhaps Tokyo's top people-watching spot, Hachiko Square, just outside of Shibuya Station, where hordes of attitude-heavy Japanese college kids and general street urchins congregate before their nightly clubhopping benders.