Rules are different in each country. We were originally told we had to give notice in the UK even though we were getting married in Vegas.
Turns out we were told wrong and we didn't have to give notice here at all. So that was £70 down the drain LOL!

I guess it makes sense. Our marriage will never appear on the registers in the UK so why would we need to give notice here?

I do, however, seem to recall the clerk at cook county courthouse saying that buy having the 'blue slip' it made her job a heck of a lot easier.

Pretty much, but there are some exceptions. You can usually find out pretty easily by doing a quick g00gle search.

Oooh, yes, good point Sarah!
My friend got married over here and whilst she and her (now) husband have EU passports, they both had to be resident here in London for 3 weeks whilst they were giving notice of marriage.

I still think it's strange that they had to give notice of marriage here as why would anyone object if they didn't know anyone?! Most odd.