Hi, I'm trying to track down a Japanese fable.  All I recall about it
is it's the story of a man who owes another man some money.  The man
who is owed the money tells the first man that he has to kill him to
make good on the debt.  The first man asks permission to go back to
say goodbye to his family.  The other man reluctantly agrees to this, and is
surprised when the first man honors his word and returns to face his
own death.  Any ideas?  Thanks.

Sounds like a variation of Schiller's ballad, "Die Bürgschaft" (The Guaranty).
One man is sentenced to death, but asks a stay of execution for a
couple of days to finish some business  - or see his family.  The
judge (?) agrees, if his friend will remain and suffer the penalty if
he does not return.  Just at sundown on the last day, he returns to
save his friend and accept his fate.

At least one version of "Beauty and the Beast" included a story
element similar to the one you've described:

"The addition of Beaumont's story, which was the basis for Cocteau's
retelling, is an intriguing touch and shows how sophisticated readers
have become since its initial release in the 18th century. It provides
story information instead of presenting pertinent conflict on stage.
The Beast is not at maximum capacity when he tells the Merchant he is
going to ruthlessly kill him, then gives the old man a three-day
furlough to say goodbye to his family. The Merchant is stupid for
intending to go back."

http://www.epinions.com/content_134456970884 

"Five Chinese Brothers" also has a somewhat similar story element:

"Five Chinese Brothers... The first brother can hold a lake in his
mouth, but a village child wanders out too far to pick up fish and
drowns when the brother releases the water.  The emperor orders him
executed by beheading, so he tells the emperor he needs to go home to
say goodbye to his family.  The second brother (who just happens to
have an iron neck) is sent in his place."

http://www.logan.com/loganberry/stump-cd.html