I'll caveat this by saying that I'm just going by ear here, and I don't have a keyboard handy to check my results. Listening to the song, I think you're pretty close. You're hearing the bass/root motion descending a third, then ascending back up by two steps. However, I think that what you're identifying as III is actually the tonic (I). If you listen to the progression, you'll hear it has a strong tendency to move towards that chord, and it feels stable just sitting there. III is a much less stable chord.
Before I go further with that thought, I want to make sure that you're aware that using upper-case roman numerals indicates major chords, while lower-case is used for minor chords. In your progression above (I II III) everything is major, which isn't how that progression would usually go. In a major key, the usual chords would be: I ii iii -- notice the minor ii and iii. (Incidentally, if you replaced either the ii or iii with a major chord, there would be a strong tendency to interpret it as a "secondary dominant" which would want to resolve to V or vi respectively.)
Returning to the song's progression, I believe that the tonic (which you had identified as III) is actually a minor chord (i). Since the progression drops two whole steps before ascending, I believe the final progression is: i bVI bVII i. For example, in the key of A minor, this would be Am F G Am. This happens to be a quite common chord progression, especially in more modal pieces, since it points to the tonic, while avoiding the dominant or the raised seventh scale degree.
One final note, if you think of this progression occurring in the relative major key, it would begin and end on vi (since vi is the relative minor). In that case, the progression would be notated IV V vi. This is the definition of a deceptive cadence.