Why should there be a rule?
I think nowadays not only it's fine either for guys or girls to take the 1st step & ask but in my line of work (barmaid) I often notice that guys seem to like when girls do the asking.
I, too, have always been more "old-fashioned" about this so I can understand you. Not that I'm overly shy but I'm quite introverted & I guess that I always prefered for guys to ask because of my submissive nature as well.
I like the idea of sending hints though. Maybe if you happen to meet someone you find potentially interesting you can then simply talk with him & find any common ground. Then if you have an activity that you both enjoy (for ex. a sport you both like to practice, or both like picnics or whatever) you could always drop the idea to that person that you'd like to share this activity with them. It's not like flat out ask for a date but you still send the message that you'd like to get to know him better & if the feeling's mutual there's a high chance he'll invite you
At least this has worked for me in the past