Chaetomorpha does not grow any holdfasts or attachment structures. Caulerpa sort of grows like a vine putting out leaves off the top of the vine and white holdfasts off the bottom, so maybe you have some Caulerpa.
here are a couple pics:
<img src="http://www.rusynyk.com/coral/Chaetocatalog1.JPG">
<img src="http://www.rusynyk.com/coral/algaetub1.JPG">