If their...whatever you call people you owe debt to accepted, say, golden pigeons or modern art to pay debt,they could pay debt in that and no one would care except the media (Breaking News! Company pays debts with buckets of paint slung at canvas! Successfully!)
The...debtees? accept dollars because they accept dollars as something of value. Granted, they'd have a hard time forcing the debtor to pay in modern art or whatever without a predefined contract, but that's not really important. The government only accepts dollars because that's all it's ever accepted; if Congress and the public all decided they wanted to, they could switch overnight from using dollars as currency to using, I dunno, purple coins with Richard Nixon's face on them with no repercussions except for economic chaos* and other countries being more certain that we're insane.
*And the economic chaos would not result from the fact that we were switching from a "good" currency to a "bad" one, but rather because of A. redrawing contracts and B. the new redistribution of wealth.
You really can't use the current economic system to justify using only certain kinds of currency; we didn't come into the economy with blank slates for minds and an infinite array of options, we came in with a decade or two of knowing no or almost no other way, with only one standard of exchange one could expect to be useable anywhere. If it wasn't for those "problems," all of these hypothetical situations could happen.
In conclusion, all that separates a worthless coin or whatever from one accepted as currency is that it's accepted as currency.