I think the chase is different for every girl, and the guy too.
To me, it used to be that you open doors, pull out her chair, always pay etc...Now when you do some of that, you'll get slapped! I've never literally been slapped, but I had a girl or two mention it to me by saying "Why did you do that? I'm an independent woman, I can open the door myself." She was kind about it, however.
I think it can be harder in today's world, in the case that you have to spend a bit more extra time seeing whether you need to do certain things that one girl may want versus something another girl may not.
Personally, I love the chase. It's humbling and confusing at times, but since I've been able to get over any type of anxiety about approaching a cute girl, it's more rewarding too. However, I don't think women should have the attitude like "I'm a girl, I shouldn't have to initiate anything."