"Neck of the woods" is a particularly odd expression to use when you live in a city and that the nearest thing to a wood around your house is the trees that line your street. However, saying "in my neck of the woods, the binmen come on Mondays" makes me feel like survival is an everyday fight in Hove. It sounds as if I lived surrounded by bears and cougars instead of seagulls and squirrels and had to get up at dawn to milk cows for my cereal instead of going downstairs to pick up the bottle that Mick the milkman has kindly left on my doorstep.
I always thought that the expression came from a time when forests were everywhere in England and people lived Robin Hood-style. But what about "neck"? It can be used in a geographical sense; it is a narrow elongated projecting strip of land (according to my dictionary), which doesn’t really fit with woods.
The Word Detective provides us with a much more exciting explanation:

    In the case of "neck," we have one of a number of terms invented by the colonists in Early America to describe the geographical features of their new home. There was, apparently, a conscious attempt made to depart from the style of place names used in England for thousands of years in favour of new "American" names. So in place of "moor," "heath," "dell," "fen" and other such Old World terms, the colonists came up with "branch," "fork," "hollow," "gap," "flat" and other descriptive terms used both as simple nouns ("We’re heading down to the hollow") and parts of proper place names ("Jones Hollow").
    "Neck" had been used in English since around 1555 to describe a narrow strip of land, usually surrounded by water, based on its resemblance to the neck of an animal. But the Americans were the first to apply "neck" to a narrow stand of woods or, more importantly, to a settlement located in a particular part of the woods. In a country then largely covered by forests, your "neck of the woods" was your home, the first American neighbourhood.

In French, this couldn’t be translated metaphorically and I’d use "chez moi", "dans mon patelin", "par chez moi".