The beauty of theories is that there are so many of them to choose from!

While TSDT is a strong harmonic progression, it's by no means the only means of organizing material harmonically. For example, other harmonic pattern include :
  • Circle of fifths (dominant->tonic G7 - C)
  • Linear (stepwise: Em - F)
  • Related chords (shared tones: Am - C)
For example, Noel's first example:

C - F - G - C

is the classic TSDT motion. But you can also see the movement F - G as a linear motion.

The substitution:

C - Dm - G - Am

creates a different harmonic movement - the Dm - G has the Dm functioning as the dominant to the G. And the C - Dm and G - Am can be seen as a linear movement.

The third example:

C - Em - F - G - Am

can also be seen as a chain of linear motion. That's not to say that TSDT isn't also present. But the Am substituting for the C in the progression weakens cadence, and makes a linear progression a better candidate for an explanation - all other things being equal (which they never are!).

It also helps to understand why particular chords are considered "dominant", and so on. For example, the vii chord is considered a dominant because it can be seen as a V7 chord without the root. In the key of C, V7 is (G B D F), and vii is (B D F). But it doesn't always function that way. Because the chord is built from a stack of minor thirds, any of the notes of the vii chord can be considered the root. So it's often used in jazz arrangements when you need to harmonize a passing note but don't want give it any particular tonality.

The relationship between the IV and ii chords are the Major/relative minor, so treating them as the subdominant is widely agreed on. But you can also relate them as to the number of notes the share - two of the three notes in the triad are shared. So it may make sense to look at the harmonic movement along those lines, too.

Treating the iii as a tonic function is something that's not seen much. More typically, it's seen as the relative minor of the V chord. So instead of grouping the iii with the tonic, it's more typically seen as part of the dominant. That is, the progression:

C - Em - G

would have the Em function as related to the G, not as an extension of the C chord.

More typically, the vi, the relative minor of I shares the tonic function.

The really cool thing is that music can often have multiple functions at the same time. For example, you can have a progression which is primarily a I - IV - V - I, but emphasize the linear motion as well, such as:

C - C/E - Dm/F - Em/G - F/A - G/B - Am/C - C

Leonard Bernstein argued that the ambiguity created by being able to have this sort of parallel function is one of the things that makes it so attractive.

-- David Cuny
My virtual singer development blog

Vocal control, you say. Never heard of it. Is that some kind of ProTools thing?