Answer: A convincing theory of chord progression is based on the overtone series: the 'harmonics' that make up every musical tone. There's a pretty lengthy discussion of these in the appendix of Exploring Theory with Practica Musica, but for the moment let's just say that every natural musical tone is a complex of pitches and that these begin with the fundamental pitch, then one of double that frequency, triple, quadruple, etc. So when you hear a single musical tone you're really hearing a number of pitches all combined. When you play a "harmonic" on a guitar or violin you're touching the string on one of the division points, stopping the fundamental but isolating one or more of the upper partials so that it's more plainly audible - but it was there all the time.

The second harmonic is the octave, and the third one sounds the fifth above that octave; subsequent harmonics grow every higher and weaker, of course. So the sound of a fifth is strongly present in the original tone, and it seems to be this that produces a sense of connection between two tones a fifth apart (or a fourth apart, which is the same thing going the other way).

Something else to consider regarding the connection between chords is the "leading tone." Play a triad on C, then one on the note a fifth higher (G) and the middle note of that second triad (B) will be only a half step away from C. This adds to the feeling that G leads to C. A similar relationship exists between F and C in the key of C: in fact, F being a fifth below C and G a fifth above it is what makes these three notes the most important in any key: the tonic, subdominant, and dominant. Chords based on those three notes occupy the three strongest positions in the key.

Next comes the chord on the second degree, because it has the same relationship to the dominant as the dominant does to the tonic - though not quite as strong because it's a minor chord and that leading tone is not there. And in general, any chord then has a pretty strong connection with the one a fourth higher or a fifth lower.

As for IV-V being strong, there are two plausible reasons: first, IV is the mirror of the V; it's the "other" dominant, the subdominant. Second, its upper two notes are the same as those of the ii chord, whose root is a fourth below V - so it shares somewhat in that connection.

Going on (and on and on), it's a good question why an ascending movement of a second makes a stronger chord progression than a descending one. One possible explanation is that an ascending second moves to a chord root that is part of the dominant of the first note. Go from C major to D minor and the D is also part of the dominant G chord; in fact its first two notes are part of the G7 dominant chord. But going the other way from D to C, C is not part of the dominant triad on D; it's only more distantly related as the seventh of the D. And as for moving from V to IV, we already know that the leading tone in the V disposes us to expect the tonic - going to IV is therefore weaker (though useful in its way; you hear V, IV, I a lot in pop and blues).

Direction also matters with root movement of a third: going up a third (C major, E minor) leaves the root of the first chord behind but seems merely to shift the root to what had been the middle note, without much change in harmony. Going down a third (C major, A minor) keeps the original root and third and adds a new root, making what seems like a more definite change. So this may be why downward progressions of a third feel stronger than upward ones.

You can make a pretty reliable set of rules for root movement (there's a version of this in Chapter IX of Exploring Theory, set up as the "Chord Progression Game.")

In order of declining strength:

Upward perfect 4th, downward perfect 5th
Downward perfect 4th, upward perfect 5th.
Upward second.
Downward third.
Downward second.
Upward third.

That's just for chords all in the same key, not adding the complication of altered tones.

Of course, one can't be strong all the time. What is strength without some contrasting weakness?