Mike Byers, a Canadian illustrator, writes a couple points for approaching Art Directors without being annoying. Some people may call persistence annoying, but here's 5 bullet points that may help to not be thought of as "annoying."
1. Target Your Audience
The absolute first step you should take is to narrow your contact list to the Art Directors who are actually looking for the type of work that you do. You’ll only be wasting their time, and yours, if you are soliciting clients who have no interest in hiring you.
Examine the work that they’ve done in the past and consider whether your work or approach fits within that scope.
Having some knowledge of an Art Director’s work is obviously a good practice anyways when building your mailing list, but it’s also the best way to make a good first impression.