Inexpensive neighboor restaurant in vicinity of theater district
What is known by word of mouth here is that rents are too high in the Loop for the inexpensive secret ethnic gem of a restaurant you are looking for. However, one does come to mind: Oasis Cafe, 21 N Wabash, located at the back of a jewelers' mall. They do delicious Middle-Eastern food at MacDonald's prices and I see that they are open until 6, which could be an early pre-theater meal (they and a lot of Loop places exist to feed office workers). Otherwise, if you can go upscale a bit, consider Russian Tea Time, 77 E Adams, or Berghoff, 17 W Adams. Check out "chicago restaurant menus" and click onto "Loop" to see the above menus. (Explanation: Loop used to be the shopping district but now is mostly business and financial. Recent effort to rehab by turning old movie theaters into legit theaters results in "theater district" in Loop, but by now most restaurants and hotels are north of the river.)
If by "theater district" you mean the area in the Loop, it is really the antithesis of "neighborhood." It's surrounded by office buildings, Mac'ys, and some recent condo projects. To get ethnic, non-chain inexpensive food that is not that far away, you could consider: (a) the places in Greektown, which are a mile to the west (but hardly a secret); (b) La Scarola, which is old school Italian, about 1-1/2 miles northwest, or (c) taking the Red Line el a few miles either south to Chinatown (again, not exactly known only by word of mouth) or north to Argyle for Vietnamese.