They don't go broody just because they've mated. They go broody only when the "mood strikes them" apparantly, or according to the "voices they hear in their head" (as other BYCers suggest).
Actually, no one really knows what exactly makes a hen go broody. It is a waiting game.
It also depends on the breed of chicken you have. Some breeds are known for their broodiness (likelihood to go broody), while others are considered non-broody (or non-sitting) breeds. I have Blue Andalusians (which are non-broody), as well as Blue Orpingtons (generally considered to be a broody breed). My 3 orp hens are just over a year old, and I'm still waiting for one of them to go broody. I've had to resort to using an incubator to hatch out some chicks, but would prefer to let the hens do it naturally. Oh well... good luck with your own "broody" issues.
Modern chicken breeds have had the broodiness bred out of them. When they go broody, egg production stops. They lay a nestfull then brood them for the survival of the species. Humans want eggs, not broody hens that stop laying eggs, so they have worked hard at stopping broody tendencies in chickens.
Certain hens of certain breeds will tend to break the bonds of modern breeding and go broody anyway, but it's got nothing to do with whether or not a rooster mates with the hen. It's all whatever her nature dictates.
You can try to select a breed that may go broody more than other ones, but there is no guarentee it will work. Here in 'Hendersons chook chart', it mentions if certain breeds tend to go broody or not, but there really isn't any hard and fast rule. Some hens will set on a nest, no matter what their breed is, some won't. ...aren't the females of all species wonderful!?!?!?!?!?
According to the Henderson chart, it appears that Delawares sometimes go broody, but the Barred Rock (same as Plymouth Rock?) are infrequent brooders.
If you want to hatch out a lot of chicks, you might want to consider getting a few broody-type hens that you can place eggs under. That was the original reason I decided to get some Orpingtons, to go along with my Andalusians, although I haven't gotten any broodies yet.