Ever notice how you don't hear 'thrice' so much when counting out things? Bob Marley didn't sing that thrice little birds were on his doorstep. I think even young Michael knew that thrice doesn't rhyme with ABC when he performed with his brothers, The Jackson 5. Although Betty White got to sneak it in to a song about Miami, she immediately got shot down by Bea Arthur (see here). It just doesn't fit in, grammatically sometimes. Yes, it's a more proper way to say three, but it's one of those words that you just can't throw into a sentance and make it sound like it goes. I'm not thrice times a lady, as The Commodores might think.
On the related note, the difference between who and whom is one of constant argument. In short (or in most cases), who replaces a noun, while whom replaces a pronoun. To whom it may concern: Who Are You by the 'orrible 'Oo (Who) never questioned For Whom The Bell Tolls by Ernest Hemingway (to my knowledge). Doesn't it sound much softer on the ears though? Try fitting thrice in that sentance and you've got yourself a jug band!
Just some thoughts of the morning. I'm open to suggestion - let me know if you can think of a better way to make thrice a normal word!