The options right around my house:
*Loi Nam Market--Small supermarket. Haven't visited this store in a while, so I'm reluctant to give an overview.
*Loi Phat Market--Small supermarket with jumbled shelves of housewares, spices, noodles, and various other dry goods. Meat and seafood counter are commensurate with the size of the market. Also, the store has a small take-out food operation near the very Southeast Asia-centric produce section.
*Phoenix Farms--Small supermarket with in-house bakery, tortilleria, and deli. Produce is generally iffy to OK. Decent meat and cheese counters, but of course, Ranch Market blows them away.
*Al-Haramine Market--Very small shop with limited selection of mostly canned and dry goods and a small cold case of dairy products (e.g., yogurt, lebneh)
*Selam Market--haven't visited, could be defunct now
Similar to Phoenix Farms in scope and size, but nowhere near my house, is Rancho Grande market in south Phoenix. Rancho Grande's meat and cheese counters are not as extensive as those at Phoenix Farms.
Baiz (already mentioned) is a great option for Halal meats, Middle Eastern staples, and fresh produce. The in-house deli is a nice place to grab a bite before or after shopping.
When I lived in Tempe, I took advantage of the student-friendly prices at Haji Baba's restaurant and occasionally would shop there for cheese, olives, and canned and dry goods. I haven't been there in a good long while, so I don't know what things are like these days.
Loi Nam Oriental Market
3838 N 19th Ave, Phoenix, AZ
Loi Phat Oriental Food Grocery
1702 W Camelback Rd # 5, Phoenix, AZ
3353 N 19th Ave, Phoenix, AZ
4418 N 19th Ave, Phoenix, AZ
4119 N 19th Ave, Phoenix, AZ
Carnicerias Rancho Grande
4227 S Central Ave, Phoenix, AZ
Two small ones not many people know about:
Village Market (Korean) @ Southern and Mill in Tempe... SE corner I believe. I adore this place. Some of the employees work at both here and the Asiana Market in Mesa.
(I can't remember the exact name) Small Japanese grocery store @ University and Priest. It's TINY but has a -couple- things you can't get other places.
re: Alice Letseat
Manila Market on the southwest corner of 35th Ave and Dunlap ( right near Romanelli's) has a good variety of Asian, Indian, even some African items. New Tokyo Market at 3435 W. Northern is a lovely Japanese market in the same neighborhood.
My all time favorite is Haji-Baba's at 1513 East Apache in Tempe, which also serves Middle Eastern food in its restaurant.
Baiz Market on 20th St south of Roosevelt. It's a really nice Middle Eastern market with an very good restaurant inside. I love their chicken shwarma sandwich.
I haven't been yet, but Shish Kebab House in Glendale was supposed to move to a new location across the street from their existing location with a market as well as restaurant inside. The move was scheduled for the end of Feb/beginning of March. The new location is the NW corner of Olive and 51st Ave.
There's a British market at Hayden/Thomas or Hayden/Osborn in Scottsdale.
There's also a small Euro/Indo market at 16th St and Bell. I can't remember the name. It's in the plaza on the SE corner.
Andreoli's is an Italian market that has a lot of fans. There's another Italian market at Scottsdale and Shea (NE corner) who's name escapes me.
523 N 20th St, Phoenix, AZ
Asiana Market in Glendale. We just visited it for the first time last weekend. It's a large Korean market with lots of prepared foods - marinated kalbi and bulgogi, kimchee, panjan, etc - a butcher counter, seafood counter, great produce section, and also a wide selection of Vietnamese, Chinese, Japanese, and Indian groceries.
4410 W Union Hills Drive, Glendale, Arizona 85308
define ethnic... there are a number of jewish/kosher markets in town now as well, and there must be tons of carnicieras around... there's an asian market that's been around a while on the west side
4208 W. Dunlop Avenue,
Phoenix, AZ 85051
and apparently another north of there..
Song Song Oriental Seafood Market
10430 North 19th Avenue,
Phoenix, AZ 85021
I have been to a good Middle Eastern Market just South of Greenway on Cave Creek (on the West side of the street).
They sell prepared foods there as well (including hot items, like Falafel sandwiches if I am not mistaken). I have never got the prepared foods, but have had good experiences getting grocery items.