Do you? Let me tell you why I do:

In 2006 I killed my buck 11/5. The rutting moon was 11/5.

In 2007 I killed my buck 11/10. The rutting moon was 10/26.

This year I killed my buck 11/14. The rutting moon was 11/13.

In the above events, allthe bucks were killed on or after the full moon. None were before. 2006 was too long ago so I don't remember when scrapes and rubs started to appear in larger numbers but I remember last year and this year clearly. Last year near the end of October there were rubs and scrapes all over the place. This year, the same thing happened right around the full moon again. Let me also add that deer were scattered all over the highways around the full moon this and last year.

I believe deer can only tell that it's time to propagate by use of temperature and available sunlight/moonlight or perhaps the gravitational pull of the moon. They don't have calendars like we do. It's the same way the salmon know to run up the rivers at specific water temperatures. I don't think that the moonis the only authorityas to when the rut begins as habitat, available food, length of seasons have a say as well, but it(the moon) certainly does seem to set an undeniable pattern, at least around here. Ithink that in states further west like Idaho, the ruthappens about 2 weeks later than the midwest but still, it follows a pattern which leads back to the moon.

Interesting reads I found on the rutting moon:
http://growthehunt.typepad.com/grow_the_hunt/2007/10/timing-the-2007.html

http://www.outdooreyes.com/phpBB2/viewtopic.php?p=295621

http://wiki.answers.com/Q/When_will_northwest_MO_deer_go_into_rut_in_2007

Before anyone tries to rip my head off, this is what I think, not what I know. If there's something you know, please share.