I recently wrote a cover letter in which I promised to follow up in a specific amount of time. My reasoning was that I would be able to demonstrate a small measure of credibility by following through on a commitment, as well as creating a setting that would side step some of the potential awkwardness of following up on applications. I should note that I did this at the behest of a friend who formerly worked in HR for a large corporation and currently teaches a university level course on all things job application.
I don't think it's a good idea.
If you're the one applying for a job, it's incumbent upon the company you're applying to to get back to you if they want to continue.
In addition, telling them "I'll write/call back on <date>" has no affect if:
a) they haven't started looking at applications by then, or
b) the appropriate person isn't available to respond on that date.
Ideally, they'd write you back right away to tell you "we won't have anything to tell you by that date; we'll contact you when we're ready", but even that requires them to parse every incoming cover letter immediately just to determine whether such a response is needed.
You will have plenty of time to show punctuality and commitment throughout the interviewing process, but this just seems like a gimmick.
Considering I've done hiring where I had to wade though 1000 resumes, I can tell you that followup is good if and only if we have had an interview. If not, then don't bother me, I will call you if I am interested. No one wants their time filled with 1000 random people calling them about a job you are not interested in interviewing them for. Hiring officials especially hate stalker types who call repeatedly. I'm not going to tell you if I am not interested, but be assured that if I never call to schedule an interview, I am not. Nor do I want to be pushed to make interview choices on your schedule not mine. There may be business reasons why we have not done the interviews yet (hiring freezes, the hiring offical having a family emergency, no resumes that make us go wow I want that guy, an overly bureaucratic HR process etc.). You gain nothing by putting that you will call in the cover letter and may actually be looked as as a potential pest to be avoided.