The first is natural and is commonly used. It suggests someone walking through a certain area, made up of several streets. The second is less natural - you can only walk on one street at a time, and we don't tend to use "on" in this context unless we want to be clear about what part of the street we are on. In BrE, we might say "I am walking in the road" to make it clear that, for some reason, we are not using the pavement (sidewalk in AmE).