As I quickly approach my due date, we have been getting lots of questions from family members along the lines of "So, are you going to call us when you're on your way to the hospital?"

Of course, we plan on calling both of our parents and will have them contact all of our siblings for us. Although I do not want many guests in our small recovery room, I understand that both of our parents and all of our siblings (12, including their SO's) will insist on visiting. I am OK with that.
However, today my mom said something like this to me: "Hon, you need to know that the word is going to get out that you are in labor. If people know what hospital you are staying at, they will come visit. They won't call you, they will just show up and there really isn't anything you can do about it."
I don't think anything else I have heard this whole pregnancy has freaked me out as much as this statement. We have at least 40 family members who live within 4 miles of the hospital, and I am terrified of them showing up! Really, I don't think anyone has a right to be there without asking first!
What the heck would you do? How do I let so many family members know that I really don't want them visiting? Did this happen to you? What did you do?