Are you sending the resume to actual advertisements or just blanketing the market? The reason I ask this is two-fold. First, if you are sending out blanket resumes, some employers may have a policy of unsolicited resumes and therefore will not keep your resume on file because of storage or EEOC reasons. Second, if you have responded to advertisements, check to see if there is a statement that says, "no phone calls."
If you do decide to call, a phrase such as, "I recently applied for the position of Pilot at your organization and I wanted to check if you received my resume. I know you are very busy. Do you have time to check that for me?" will work fine. I would NOT start out by stating my name, because if they get annoyed with you for calling, then you have not provided your name for them to remember! If they say that they will check, THEN provide your name. If they say no, they do not check, thank them for their time.
However, I really tell my clients not to check up on a submitted resume, only about checking on the status of the decisicion after an interview. The HR people are usually swamped and do not have time to check on submitted applications or resumes.
Just my two cents,
Kathy