If you live in the area, then I would highly recommend visiting Mekong Supermarket especially for your fresh vegetables and fruits purchases.
That said, it's not my favorite Arizona Asian supermarket.
Mekong is a nice sized supermarket with a large seafood and meat counter. Dry and canned goods are plentiful. The cashiers are young, friendly, and fast. If you want to avoid the crowds, then go on weekdays. The free ice and bags after the check out lines are much appreciated and will help keep your meats and seafood cold until you get home.
There are several restaurants and shops in the same plaza, so you can make a day of it if you want.