What I normally do is call Marriott Reservations to get the hotel GM's email address. I will then email the GM to let her/him know that I will be traveling in with my family. Give the name, dates, reservation info. I then let the GM know that I have heard many good things about the hotel and we are really looking forward to our stay. (Stroking their egos a little bit usually doesn't hurt.

) Then I let them know that I understand no upgrade is guaranteed, but anything they can do to help us out would be greatly appreciated. If the hotel isn't too booked up, this usually works for a decent upgrade. At most hotels, the GM will forward the email on to whomever on staff who assigns rooms.
It might not hurt to call either the day before or the morning of your arrival and remind them that you are coming. You can inquire as to your room upgrade then. You can also ask about an upgrade when you check-in as well.