Should I ask to meet graduate students during a TT job interview?
There is two different styles of "prospective faculty meets grad students" meetings:


The interview (the grad students are in some form on the selection panel or their input is otherwise valued by the panel)
The applicant gathering data (the grad students are not involved in the decision, but the applicant is interested in getting a different - probably more accurate - perspective on how the department really operates)


These are fairly different meetings. The former is basically the same as all the interviews with faculty, only that it's now a grad student on the other side. The latter should be much more driven by the candidate, and it's the candidate who will ask most of the questions. 

For the former, the department will tell you whether you should meet students, and which. If they do not have it planned and you don't care about the second angle, there really is no point in meeting with students.

However, I have found the second angle to be extremely valuable if you have any doubts at all about the institution. Grad students tend to be more honest about administrative procedures, quality of life, work culture, etc. However, you need to enter this with the right mind set - this is not part of the interview, but people external to the entire process doing you, the candidate, a favor. It pays to be friendly and open here, and putting down the interview guard just a little may be necessary to get an open conversation going. I have found a good pattern to be to take out 2 or 3 grad student volunteers to an informal lunch (if schedule permits), or at least to a longer coffee break - and the candidate should invite them (or, in some cases, the department offered to pay, which is also fine as long as the students are invited).


  I worry that, if this is not standard, then it might be an awkward request for them. One school of thought is that this is a bad sign I should be aware of, but I don't think I should read too much into it.


It may be a little non-standard, but not standard is good! Your goal isn't to appear exactly like the other four candidates. If your request makes you seem more interested, more thorough, or just plain nicer than the others, this is a good thing. I am not sure what "school of thought" you are talking about, but trying to be as uniform in your requests to any other candidate seems horrendously bad advice to me. That is not to say that every department will say yes. Sometimes they really don't bother setting it up, sometimes they simply did not find students that were interested in meeting prospective faculty, both of which may be a small red flags on their own.