My advice to the OP is to mine the substantial, nay stupefying wealth of prior posts here on Pilsen, Taylor, Tri-Taylor, Oakley, Chinatown, Bridgeport even the South Loop, then come back at us with a more detailed request.
I mean, really. This ain't that other board.
I live in the area (along with probably several hundred thousand other people) mentioned by the o.p., Hikari, and write regularly about places in this part -- nay, these parts -- of the city; in terms of quasi-official nomenclature, the zone mentioned includes presumably all of the Near West Side and at least much of the Near Southwest Side and, insofar as the West Loop isn't just part of the Near West, then a bit of the West Loop as well; neighbourhoodwise that includes Greek Town and the stretch of territory extending to the west, out I suppose to whatever they call the area around where CrazyC lives, and then to the south of the highway (Eisenhower) also Taylor Street, Tri-Taylor, Pilsen and
Heart of Chicago. And adding in the South Loop (as being 'near' UIC), as JeffB suggests, isn't stretching things too far... (I'm not so sure though about Bridgeport)...
Anyway, I tried writing a response twice yesterday and ended up scrapping both because they sounded too cranky. The fact is, as JeffB points out, a really broad request of the sort made hereabove should probably be narrowed down a bit through use of the search-engine of the site. Once the request is focussed a bit more, answering it seems to be no longer an open-ended chore but rather a doable and gladly done task in which one gets to share one's knowledge.
P.S. OK, sorry about all that. If you are completely unfamiliar with the area, you might try Nuevo Leon in Pilsen for starters. A rare combination of quality and general popularity. Same for Bruna's on Oakley. Let's say Tufano's and Gennaro's on (or near) Taylor, though nothing on Taylor gets me too excited, save Conte di Savoia, Mario's, and yes, Al's, though none fit the quiet dinner requirement.
You know you'll be awfully close to Greektown as well.
The above are (probably) all appropriate suggestations, though the vagueness of the original request leaves one in some doubt. To that list, albeit with lingering doubts, one might add Artopolis on Halsted between Jackson and Van Buren, Wishbone at the corner of Washington and Morgan, the Palace Grill (a diner) at Loomis and Madison, Cous Cous (Moroccan) at Taylor and Bishop, Polk and Western Hot Dogs (:o *) at, that's right, Polk and Western.... There are many good, inexpensive and casual places in this fifth of the city in which to eat without having to resort to chains.
Yeah, the request ought to be a bit more focussed... Are the suggestions just to be for general reference or for some particular occasion (e.g., before or after an event at the United Center)? Give us a little more info and I promise you'll get some good suggestions.
* I mention this place only because it was cited somewhere on the board in a post yesterday as being among the best purveyors of hot dogs in the city. I have no opinion on the matter, having not yet eaten there, despite the fact that I pass within smelling distance of the place almost every day -- they need to change the grease in the deep-frier more often, in my humble opinion.