Eixample is alright, and supposedly a decent place to live, but the main touristy part of Barcelona is the old town, Barri Gòtic, which is a typical chaotic maze of streets. Eixample receives a lot of visitors too, of course, but not so much, I think, because it's very charming in itself, but because it makes up a large part of the inner city and thus contains a lot of tourist attractions (including Sagrada Familia and other buildings by Gaudí) and shopping streets.

Also, once you go beyond Eixample (except to the east) the city's not a regular grid anymore. In the end, Eixample and other, newer neighbourhoods built on a grid probably represent a (substantial, of course) minority of the city.