Great burgers and sandwiches at the places mentioned above, but you might never feel that the price is "right" and the side orders will be smaller than you are used to (but remember, this is healthier.... ).

I think RichardInSF is thinking of Kua'Aina, which has expanded its branches in the Tokyo area. They do a pretty good burger and fries:
http://www.kua-aina.com/

And I will also add three places to this list.

Mos' C - in Kichijoji and Kagurazaka.

OH DEAR - SCRATCH THIS RECOMMENDATION
I've had two mediocre experiences at Mos' C (now renamed Mos' Classic). Distinctly lower quality than before. Sorry to say that this place no longer makes a great burger.

This place is not your father's MosBurger.
It is a high-end burger outlet operated by the Japanese MosBurger chain. The burgers and sandwiches themselves are superb. But there are some Japanese quirks about the place - for example, they only give you three or four french fries and a tiny pickle with your order. On the other hand, they offer much better selection of beers than I usually find in America.
Prices are high. (Two people won't get much change from 5000 yen after burgers, side salad, order of onion rings and a couple of beers).

The Exchange - part of the Palace Hotel (Otemachi)
Excellent delicatessen-style sandwiches - and they should be at the prices they charge (about 1300yen for an American Club S'wich). This is the place to come if you want a "pastrami-on-rye" or a "BLT-on-French-go-easy-on-the-mayo". They offer a pretty good dill pickle with the sandwiches, but no coleslaw or french fries. (Shame). Don't even think of buying a drink in this place (400yen for a can of soda - almost four times what you pay in the convenience store in Otemachi station, 50 meters away).

Wolfgang Puck Cafe Opening all over Tokyo. Better than average burger and fries.

So, you should have no problem getting your fix of America among Tokyo's "cosmopolitan," dining culture....

But how about the idea that Tokyo is "cosmopolitan?"

Well it really isnt' a "cosmopolitan" city, in the sense of having many pockets of predominantly foreign populations, such as you might find in the "Salad Bowl" cities like Los Angeles, London, New York, etc. I guess Tokyo just has fewer people from overseas than those places, and even fewer of those overseas arrivals become long-term residents.

As other people have mentioned, there are some well established, long-term communities from Asian countries: There are "Little Koreas" in the Okubo area (near Shinjuku) and around Ueno. There are also several vibrant Chinese communities. If you go as far as Yokohama, there's even a "picture post-card" China Town with dragon gates and everything.

There is a kind of "Quartier Francais" in the Iidabashi area, which stands out as a rarity - in that one Western nation has put its stamp on the neighbourhood. Now don't get me wrong - This is not an area where you can communicate in French. Most of the people you will see around Iidabashi are Japanese, but there are enough French people living here to make a cultural impact. I guess it might have started because parents wanted to live near the French school and some French people were working at the L'institut Franco Japonais de Tokyo, and then the supermarkets started stocking products from France...
Finally, there are now many French restaurants in this area - but most of them are run by Japanese people - offering their take on French cuisine (some of which is excellent).
Once or twice, I have been asked by store owners in this area "Which part of France are you from?" I suppose that many of the Westerners that they come across in the neighbourhood are from France. (I have to tell them that I'm not from any part of France...).

As for a "Little America" district in Tokyo. Well, there is the "American Club," which acts as a haven for the long term ex-pats ... but that's a rather exclusive, member's only "Little America."

Someone jokingly mentioned "Amerika-mura" in Osaka. How about "Ame-yoko-cho" in Tokyo? (I believe it got its name during the American occupation of Japan after WWII, because the black market there was selling loads of products that had been "liberated" from the American military stores). By all means take a look - but don't go there expecting anything American. It's a quintessentially Japanese experience (and a nostalgic trip down memory lane to the austerity of post-war Japan).