For everyone who thinks (like MtheGM) that I'm overcomplicating something simple, all I can say is that it looks different when you come at it from the outside. I can work out tunes by sol-fa, turn them into notes & transpose them between keys, but chords are a closed book to me. (I can play two chords on the guitar - one of them's G and the other one isn't.)

This all started when I wanted to put some chords to the melody of a song, with a view to putting together a very basic melodeon accompaniment. I knew which bit of the melody the chord should go under & even which note of the melody it should support, but what chord should it be? With a bit of trial and error I discovered that the chord that sounded right would always have the relevant melody note in it somewhere, but the 'right' 1-3-5 chord might have that note in any of the three positions. (In other words, if the melody note was an A and the tune was in C, the 'right' chord might be D-F-A (D minor), F-A-C (F major) or A-C-E (A minor).)

But what this suggested to me - after I'd done a bit more experimenting and made a lot more notes - was that the 'right' chord for any note, given a tune in the key of C, could only be C, F or G major, or else A, D or E minor. This made me feel I must have gone wrong somewhere, as I know I've seen tunes written out with a much broader range of chords.

So I'm really interested by Anahata's comment -

bear in mind that the home key of a song or tune is only the key it starts and finishes in (usually). On the journey from one end to the other, it may well wander off into other keys, even if only very temporarily.

although I suspect it raises more questions than it answers. I've only played 50-odd tunes, but out of those I can count the ones with accidentals on the fingers of one hand - and almost all of those are flattened sevenths, generally sharpened again the next time you come to them (a la Elsie Marley - those half-Mixolydian tunes remind me of a stair tread that sometimes creaks when you step on it and sometimes doesn't). Some of the chord sequences I've seen seem to range much more widely. I guess the question is whether accidentals in chords are much more common than in melody lines - and whether you generally have more latitude, in 'chording' a melody, than the three-chord trick would imply.