I take something of a mechanical, deductive approach, which is arguably not ideal compared to listening for tension, resolution, and chord quality (though those things can absolutely still be part of it), and is pretty time-consuming, but it's how my brain works. Take it or leave it as suits you.

First, I find a note in each chord. Whatever note you can pick out. This gives me a sequence of notes. I match this on my electric piano that's right next to my computer. I now have a sequence of intervals.
 
For instance, let's say I match the notes B, C#, D. The interval pattern is +M2, +m2. Let's say the choices are ii-V-I, ii-I-V, and ii-IV-I. Which of those have that interval pattern?
 
I then refer to a cheat-sheet I have of the notes in the chords in the key of C. Again, it doesn't matter whether the progression is actually in C, as I'm just looking for interval patterns.
 
Does ii-V have an interval of +M2? Yes, taking the key of C as an example, Dm (DFA) to G (GBD) has two of them, A to B and F to G. What about ii-I? Dm to C (CEG) has two as well, D to E and F to G. And ii-IV? No, Dm to F (FAC) has no such intervals.
 
OK, so now we've eliminated one of the possibilities. To narrow it further, let's look at the second interval in the pattern. So, if we've gone from Dm to G (ii-V), we've landed on either the B or the G according to our interval pattern (relative to the key of C -- of course, our absolute notes are different). Does the I chord (to complete the ii-V-I pattern) have a note that is a m2 up from either B or G? Yes, the root of the C chord is up a m2 from the B in the G chord. So, that's a match.
 
But is the other pattern a match, too? If we go from Dm to C (ii-I), we've landed on either the E or the G (again, relative to the key of C). Does the V chord (to complete the ii-I-V pattern) have a note that is a m2 up from either E or G? No, it has the notes GBD, none of which fit the bill.
 
Voila! We now know that ii-V-I is the only match among the choices.
 
We've traced through the notes A, B, and C as matching the interval pattern, but of course the actual notes we matched are B, C#, D. We can now surmise that the progression is actually in the key of D (with chords Em, A, D), but the game doesn't require us to identify the absolute chords, just their relative harmonic functions, so that's just a bonus.

I'll add that sometimes just recognizing and matching the chord quality -- major, minor, diminished -- is quite sufficient to narrow down to the right answer. And if it doesn't narrow it all the way down, it can at least eliminate some of the possibilities to speed up the above process.