There must be some polite way to see someway and say, "Oh hey! I know you tend to be busy and not at all deadline-oriented, and I was just wondering if you'd written those letters of reference that are due this week, or even remembered that you agreed to write them".
There must be, but I don't know what it is. So I just stalked this particular professor at his office hours, a week after those deadlines had passed, on the pretext of asking how he would like to receive the next list of jobs to which I would be applying, in the hope that he would in some way indicate whether the earlier batch of letters had been sent. This turned into a 45-minute conversation on the benefits of microwave coking, his work with the Teamsters Union, and our mutual preference for CDs over mp3s. Buried within this array of topics was the information that he did remember the letters, and had kept the folder of supporting materials I provided in a safe place as he rearranged his office, but had not remembered the deadlines, or written anything.
In my gentle prodding way, I volunteered myself to produce stamped, addressed envelopes, with departmental letterhead, and a quick checklist of colleges and addressees first thing the next morning. I am deliberately not worrying about how an incomplete application will affect the opinion of the search committee, which only affects three jobs, two of which I wasn't excited about. I will, however, be stalking this professor at his office hours again later this week, or contriving a "coincidental" meeting in the hall where he might spontaneously update me on his progress.
I don't mind my job prospects depending on other people, in the form of the search committees that will make the employment decisions. I do mind my job prospects depending on other people, in the form of other people sticking to deadlines that really only matter to me.