Just as a quick caution, there are certainly ethnic neighborhoods in Chicago, but I wouldn't limit myself to only trying ethnic food in one neighborhood. Quantity doesn't always equal quality as the neighborhoods have grown and changed. For example, Ukranian Village was mentioned, and it's now mainly known as a collection of hip, trendy bars and restaurants rather than Eastern European. Lincoln Square used to be a German neighborhood but now only has a couple authentic German restaurants left.
That said, one would be missing out if you don't try Mexican in Pilsen or Vietnamese on Argyle St. But there are lots of good ethnic restaurants scattered throughout the city.