As a complete food addict, and avid culinary traveler, I'm always (always!) looking for new places to go, new things to taste, and overall checking out the best places from around the country.
So where are your favorite local places to go? And in the interest of at least my own personal use, I'm going to use this post as a place to keep all the recommendations indexed. Location, Cost, and any Must Haves.

The more unique, the better!
I'll try to edit all the recommendations into this post if I can, let's see how this works out.

I'll start out with the places I can think of.
(Cost scale: $ cheap eats, $$ casual eating, $$$ nicer sort of place, $$$$ good and expensive)
Alabama
Iguana Grill
Location: Birmingham, AL
Cost: $$
Gotta get: It doesn't matter, the food is good but not the main appeal.
This is the most beautiful restaurant I've ever seen. I would pay just to sit and look at the decorations.
Washington
Dic's Drive In
Location: Seattle, WA
Cost: $
Gotta Get: Dic's Deluxe (1/4 lb burger)
Seattle's classic drive-in, get burgers, fries, and old fashioned shakes.
Bizzarro Italian Cafe
Location: Seattle, WA
Cost: $$$
Gotta Get: "Forest Floor Frenzy" - Mushrooms and parpedelle.
A small, quirky Italian restaurant in the neighborhoods of North Seattle. Great hand-made pastas and always an adventure.
Saigon Bistro
Location: Seattle, WA
Cost: $$
Gotta Get: Banh Xeo - Vietnamese Meat-filled Crepe
In the busy heart of Seattle's Little Saigon is this authentic Vietnamese lunch spot, serving traditional dishes that you won't find at any pho joint.
Cedar's Mediterranean
Location: Seattle, WA
Cost: $$
Gotta Get: Butter Chicken
This favorite sit-down spot for UW students has a wide selection of Indian curries as well as tandoori cuisine. Try the gyro, too.
House of Hung - New Hong Kong Restaurant - Jade Garden
Location: Seattle, WA
Cost: $$
Gotta Get: Dim Sum
Dim Sum, the Chinese meal of hot tea and small steamed dishes of seafood and meats is always a favorite around here for a fun group meal. This trio of restaurants in Seattle's Chinatown all compete for the weekend lunch crowds, but any of them is sure to give a good meal.
Red Mill Burgers
Location: Seattle, WA
Cost: $
Gotta Get: Verde Burger and Onion Rings
Amazing burgers at a great price. They also sure know how to make bacon. Killer onion rings, and a roasted chili "verde" burger.
Spiro's Greek Island
Location: Kent, WA
Cost: $$
Gotta Get: Huge Lunch Gyro Sandwiches
Hidden in the suburb of Kent is a great Greek dining spot.
Tat's Delicatessen
Location: Seattle, WA
Cost: $
Gotta Get: The Pastrami
A favorite Seattle lunch spot, serving fresh New York-style deli subs on fresh baked breads.
Salumi
Location: Seattle, WA
Cost: $$
Gotta Get: The Meatball
Known for its authentic artisan cured meats, Salumi serves up as close to what a sandwich in the Tuscan countryside would be like in the heart of Seattle.
Pho T&N
Location: Poulsbo, WA
Cost: $$
Gotta get: Stir fried beef with lemon grass, beef pho, Vietnamese iced coffee
Somehow in one of the whitest towns in WA, there's an absolutely fabulous pho restaurant, and so far EVERYTHING on the menu has been awesome, but I'm addicted to the beef with lemongrass.
Ezell's Chicken
Location: Edmonds & Seattle, WA
Cost: $
Must have: Spicy fried chicken, various sides (who cares, it's all about the chicken really)
Best fried chicken in the area, far superior to the kfc/popeyes of the land, and the spicy is not too hot but way better than the regular IMO.
OK Corral
Location: Seattle, WA (Greenwood)
Cost: $$
Must haves: hush puppies, brisket, ribs, hotlinks
Honestly, you can't get anything bad here, and it's not that expensive for BBQ. As always, if you get hot links, you've got to have some collard greens, and everything is made from scratch, one of my top 5 in the PNW.
Gorditas
Location: Seattle, WA (Greenwood)
Cost: $
Gotta get: mondo burrito.
Their mondo burrito is massive and delicious, I used to take 2/3 of it home and have two extra meals out of it for ~$10, that's good & cheap eats.
India Mahal
Location: University Place, WA (Tacoma area)
Cost: $$
Gotta Get: Tikka Masala, Naan, anything with lamb
Absolutely awesome curries, wonderful rice, and the grilled breads are amazingly tender & delicious
Louisiana
ANYWHERE!
Location: New Orleans, LA
If this doesn't work, blame Nate.

(Daniel's addendum: "But you should be warned that Jager Haus is unbelievably slow.

")
Missouri
Tucker's Place
City: St. Louis, MO
Cost: $$
Gotta Get: Steak Specials
Best steaks around for a way-more-than-fair price. Steaks come with huge potatoes, or try the lunch combo for a great soup and sandwich.
Arthur Bryant's
City: Kansas City, MO
Cost: $$
Gotta Get: Pulled Pork or Ribs
The definitive Kansas City BBQ spot, they make it, and make it good.
El-Rodeo Mexican Restaurant
Location: Rolla, MO & Bolivar, MO
Cost: $
Gotta Get: Everything!
These people have such great food at such amazing prices. Their cheese enchilada's are pretty dang good.
Texas
Buzzard Billy's
Location: Waco, Tx
Cost $
Gotta get: Try the alligator, I'm from Kansas, we don't get many chances to try alligator!
Tennessee
Ri'chard's Cafe
City: Nashville, TN
Cost: $$
Gotta Get: Beignets!
Friendly cajun food in a Nashville hideout. Live music, too.
Noshville
City: Nashville, TN
Cost: $$
Gotta Get: The Reuben
Classic deli-style fare in a laid-back atmosphere. Sandwiches are big and delicious.
Las Paletas
City: Nashville, TN
Cost: $
Outside of downtown, in a small corner with no sign, Las Paletas serves up the best popsicles you've ever had in your life. No joke.
Marche
City: Nashville, TN
Cost: $$
A classy, open cafe downtown. Seasonal chef-chosen Euro-inspired favorites round out the menu.
Georgia
Shirley's Soul Food Cafe
Location: Toccoa, GA
Cost: $
Gotta get: Just about anything
Just good southern food. Menu often depends on what they caught while fishing but usually they have Fried Catfish, Fried chicken, Okra, Frogs Legs, Cream Corn, Collards, Sweet Tea, Ect. Usually a good idea to not eat anything all day and then eat dinner there.
Flip Burger Boutique
City: Atlanta, GA
Cost: $$
Burgers with a gourmet twist. Strange, yet fascinating. Try a liquid nitrogen milkshake while you're at it.
Gladys Knight & Ron Winan's Chicken and Waffles
City: Atlanta, GA
Cost: $$
Long lines, but really, really good fried chicken and other assorted comfort foods.