flyby with a "Hello, Tower"
A step "closer" than just a discreet flyby, this strategy DOES involve actually stopping in to the home/establishment where you wish to unobtrusively check on the conditions/progress of something relating to you, but you still do not actually inquire about said situation; you merely "exchange pleasantries and make small talk" with the people there about the weather or whatever, and then quietly leave again after a minute or two. The theory here is that if there is any problem/delay/question regarding what you are wanting done, the people will likely tell/ask you about it while you're there... they know fully well why you've dropped by --- you are wondering how things are proceeding --- and so there should be no need to make anyone feel pressured or irritated by actually alluding to the subject.
The local garage is often "swamped up to their eyeballs" with the myriad of vehicles brought in for repair, and so it is often a month or more before they are able to get around to addressing a car that I bring in. They do always eventually repair my vehicle, though, so I seldom actually ask them about how soon they can get to it; I simply do a tentative flyby with a "Hello, Tower" every couple weeks, just in case they need to speak with me regarding any problem/holdup with the repairs.