Those which I caught, I threw away. Those which I could not catch, I kept.
The answer is one word.

Hello knowitall22-ga,


Thanks for the fun question.

Those which I caught, I threw away. Those which I could not catch, I
kept.  The answer is one word.

The answer to your riddle is lice.

I found the answer to this riddle at the enigma graphics website. 

Riddle number 7.

“What we caught we threw away; what we didn't catch, we kept.
--the riddle put to Homer by fishermen of Ios, said to have caused his
death.”
http://www.enigmagraphics.com/riddles/

"What were the fishermen of Ios speaking of when they told Homer: What
we caught we threw away, what we didn't catch, we kept.
His intelligent response was:
Lice, I have lice myself." 
http://www.geocities.com/SouthBeach/1732/riddlew.htm


This riddle is included at the answer bank as well.
http://www.theanswerbank.co.uk/DisplayAnswers.go;jsessionid=aaa2oGgew1oMQ8POfSt?question_id=22760&category_id=12&index=3


Search Criteria:
Those which I caught, I threw away


I hope you find this helpful and if there is anything that I've
written that needs clarification, please ask before you rate this
answer.

Best Regards,
Bobbie7-ga

Thanks, bobbie7...you are correct.