The simplest way to make a large area non-symmetrical is to draw one half (third, quarter, segment, etc.) and then cover it over when you draw the next. I have a friend who is very much into symmetrical dungeons and maps. Looking at the whole thing at once causes him to go back and rework things until it hits his idea of exact symmetry. I just focus on a little area at a time and I tend to get non-symmetric, wandering things without much sense to them.

Another option is to use a random generator to lay out basics and adjust details (or use random elaboration on a very high-level plan). A random generator might be something like a set of geomorph cards or a piece of software.

The final option is to go for places that would naturally be symmetric, like grand public spaces (churches tend to hit this nicely). Then elaborate little bits here and there to break symmetry. Work your way up to the less symmetric.