Ideal neighborhood for non-meat foodie
Well, the problem is when I think of the great vegetarian places in Chicago they are not confined to one neighborhood. I'll let fish oriented places get covered by others and I'll toss my vegetarian suggestions to you:
For example, Blind Faith is a great place to try - but that's in Evanston (north of Chicago). http://www.blindfaithcafe.com/ You actually will find quite a few decent places in Evanston, as well as shops, an artsy movie theater, and an array of other things. Chef's Station would certainly be worth investigating: http://chefs-station.com/
Soul Vegetarian East is a place I'd say is a must for vegetarians - that's on the south side, 75th street. http://www.kingdomofyah.com/SV.htm There isn't a whole lot else around here for vegetarians.
Russian Tea Time is a place I always recommend for vegetarian faire, that's in the Loop. http://www.russianteatime.com Next to it is Rhapsody which I've had tremendous meals at: http://www.rhapsodychicago.com/ Park Grill has some options for you as well: http://www.parkgrillchicago.com/ This is not really a "neighborhood" - you'll be near Grant Park, museums, etc. Roughly in this area is Atwood Cafe: http://www.atwoodcafe.com/ If you're staying in the Loop, you'll be set.
While North Michigan Avenue has options, the only one I know of and recommend is Cafe Spiaggia which I found to be phenomenal: http://www.levyrestaurants.com/NR/exeres/68A016EE-9274-4602-AB13-975351735BED.htm
The row of Indian restaurants on Devon is a must as well, but this is near nothing else I mentioned: http://www.devonavenue.com/freebusinesses/restaurants.htm I have been to about six of the places on Devon and Tiffin was my favorite.
Karyn's Raw or Karyn's Cooked would be worth a look: http://www.karynraw.com/
Chicago Diner is good if you want "mock meat" - there isn't much attempt at creative vegetarian food, it's all fake pork, fake beef, fake chicken, etc. (Thats not a criticism, but I know some vegs aren't into that type of food) http://www.veggiediner.com/
Green Zebra is the high end vegetarian place in Chicago and it's small plates: http://www.greenzebrachicago.com/ For fish, their sister restaurant Spring gets good reviews, but I've never been: http://www.springrestaurant.net/
I enjoy Earwax in Wicker Park: http://www.earwaxcafe.com/
My personal favorite restaurant in Chicago which gets bashed here for various reasons is Frontera Grill. High end Mexican, owned by celebrity chef Rick Bayless, fantastic food, but no reservations means you'll be waiting as long as three hours on a weekend. My advice is to go at an "off" time - either get there 20 minutes before they open or 9 pm. http://www.fronterakitchens.com/restaurants/restaurants.html
I don't care for Greek food in general, but Greektown always gets recommended for veg/fish eating types: http://www.chicagogreektown.com/
Thank you for the Russian Tea Time recommendation. My SO and I were in Chicago last weekend (visiting from Minneapolis) and went to RTT based on this post. The vegetarian platter for 2 was HEAVENLY.
Every bite was so good, and there was SO much food. It came in two courses - an appetizer plate with potato dumplings, stuffed mushrooms, beet caviar, tabbouli, hummus, carrot salad, beets with capers (accompanied by delicious pumpernickel bread), then a main platter with chickpea stew (wonderful flavor, laced with cinnamon), mung bean stew, stuffed green pepper, stuffed eggplant, and a layered dish of potatoes, cabbage, and veggies.
We came with a big appetite, but couldn't finish it all. I would highly recommend this, and I will definitely return on future visits to Chicago.