I'm linking to this thread from the Starwood forum as FTers are missing out on a good dining area when staying at the Sheraton Miyako Hotel.
Shirogane Shiroganedai Platinum Street dining area guide Gaien Nishi Boulevard

A few steps from the Miyako - There's Enju, within Happoen Garden, 1-1-1 Shirokanedai (Enju English website)
Al Ceppo, 3280-5005, Shirokane 2-3-19 set lunch from ¥1,015 - includes coffee (dessert ¥350 extra) MAP
Takanawa Samrat Curry House 5420-9230, Takanawa 1-2-1 BUDGET OPTION Located to the North of Shirokanetakanawa station MAP
Pizzeria Salvatore Cuomo(English), 5447-6465, Shirokane 1-29-9 located just to the North of Shirokane Takanawa Station MAP (associated article) Pizzas can be delivered
The Toei 94 bus runs from the East of the Miyako to this area (¥200 - but it's only a couple of stops... don't overshoot! See route details below)

Or for v. good Ramen you could walk South to Takanawadai station
メンタツ Mentatsu Ramen 3443-6659, Shirokanedai 2-26-13 Map Here- Open: 11：30〜15：00 & 18：00〜22：00
The 94 bus runs here if you don't feel like walking back

Restaurant & Wine Celler Davis English review 3440-6007, Takanawa 2-5-6 MAP
starting from the Miyako, walk South along the orange road (Sakurada Dori) - at the AMPM convenience store turn left (East) which will lead you onto a street just South of the post office - Davis is around here
Manuel Churrascaria 3443-5002, Takanawa 2-3-22 English review MAP


If you're Kosher (see post#3) or like Israeli food there's David's Deli (click for review) just to the North East from Shirokane Takanawa Station on Sakurada Dori (orange road), 5-13-13 Mita (Mita 5 - top right of this map)


I'd advise anyone after a modestly priced meal to walk Northwards to the Shirokane 3 area. Parallel to the river/shuto expressway there are streets with plenty of bar and restaurant choices. MAP of area
eg: Labyrinthe (French), 5420-3584, Shirokane 3-2-7 (near the post office) - prix fixe lunch approx ¥1k, dinner considerably more
For inexpensive but good food in the evenings there is this yakiniku place:
Suzukiya, Shirokane 3-9-8 MAP

If you stray in to the Shirokane 1 area you should also find a few restaurants there too
壱参らーめん Ichisan Ramen, 3280-2237, Shirokane 1-3-14 (open until midnight)Map
EDIT TO ADD - this is definitely an up and coming area, more and more good restaurants are opening up around here. A trio of just opened galleries might also help.

Just to the West of Shirokane 3 there's Pizzeria Isola (English), 6-17-2 Shirokane
or
Chez Tomo Shirokane 5-15-5 (Food photos here) Lunch: 4 courses ¥2,890 - Dinner: 6 courses ¥5,780 - Organic Vegetable menu: 5 courses ¥5,780 (bottle of wine ¥5k) - NOTE this place now has a Michelin star
シロガネ Shirogane 3449-0033 3rd Floor, 6-16-28 Shirokane (above the Mercedes Benz dealership) for a more formal Japanese meal lunch from ¥2k to ¥10k, dinner ¥6k upwards, Hanagoromo course ¥3k or
甚六 Jinroku (English) 3441-1436, Shirokane 6-23-2
心米 Kokoromai 5793-4556, Shirokane 6-18-7 2F English Review - Yomiuri Review JP Recommended for rice connoisseurs set dinner menus from ¥4k
(This area is near the Northern end of Platinum Avenue)


And of course, there are plenty of upscale choices on Platinum Avenue itself (also known as Gaien Nishi Dori) just to the East of Shirokanedai station such as
Blue Point 4,19,19 Shirokanedai
Sakura Sakura (Engrish), 5-15-10 Shiroganedai (tucked off the avenue on the other side to Blue Point - map).
Il Grappolo is another option, 4-9-18 Shirokanedai (basement) (food photos)
There's also a
La Boheme (English) 4-19-17 Shirokanedai (part of a popular, and moderately priced, Italian style chain)
Or for that Michelin 3* blow out:
Quintessence - MAP

Platinum Avenue is the yellow road running North to South.

Here's a list of just some of the restaurants on Platinum Avenue (purachina dori):

Japanese
茶茶 ChaCha, 5792-5792, Shirokanedai 5-14-1 evening set menu from ¥3,675 and an unlimited 2 hour 'bar for ¥2,1000 (URL)
ひづき Hizuki, 5798-7492, Shirokanedai 5-11-2 (English)
光仙 Kosen, 5447-8255, Shirokanedai 5-17-11
三合庵, Sangouan, 3444-3570, Shirokanedai 5-10-10
利庵, Tosian, 3444-1741, Shirokanedai 5-17-2
*(fusion) 雫 (シズク) Shizuku, 5449-4977, Shirokanedai 5-17-10, B1 (Japanese article)

French
L'ecailler, 6408-0879, Shirokanedai 5-3-8 3 course pasta/rice lunch w/ coffee from ¥1.8k
Ozawa, 3442-1171, Shirokanedai 4-9-23 (food photos)

Italian:
Alberato, The House Shirogane, 3449-2662, Shirokanedai 4-9-19 (URL in left column - Menu HERE jp)
Ristorante Luxor (English), 3446-6900, Shirokanedai 5-4-7 pasta & salad/soup lunch from ¥1.5k + ¥300 for dessert
*(fusion) Stellato (English), 3442-5588, Shirokanedai 4-19-17 (food photos)

Bar Tsuya Shirokanedai 5-2-5 1F

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Within the Sheraton Miyako itself is the Cafe California.
Not the most economical of options, this is what ¥1,785 will buy you at lunchtime - note, the hamburger is ONLY available at lunchtime. The reviewer did say it was a good one, even though he blanched at the price!


Here's the Google Satellite Map, centred on the Miyako.


And if these aren't enough options, you could just get the 94 Toei bus on Sakura Dori (which runs along the East of the Miyako) Northwards to Azabu Juban which is great for restaurants. Trip costs ¥200 and takes less than 5 minutes. And Azabu Juban is a 30-40 minute walk away (or just a few hundred ¥ in a taxi) so you wouldn't get stranded.
Here's a great example:
Yukimura Azabu MAP great but expensive
or there's the more affordable Kan Nichi Kan (MAP) for excellent Korean food - EDIT, big change in management and restaurant is not nearly so good anymore


Vegetarian?
Get the shuttle bus to Meguro, from there walk to
http://www.woodmoon.jp/ - MAP

A little further afield, but perhaps worth it if you want to eat more swanky vegetarian food
Merleau Pannonica - best to make a reservation before turning up as there are only 17 seats
(Meguro Ward) Aobadai 2-10-11 Saijo Mt. space 1F
Tel: 03-3464-0100 , open:18:00 - 22:30
MAP