There are a few different ways to go about such things. Here's one:
Open the file, and choose File > Save As to save it under a different name in a different location. This will keep links intact so long as the original linked files stay put.
If storage space isn't a concern, you could replicate the linked file set using File > Package to save copies of the file and all its linked files in a new folder. There is also a way to copy-and-migrate the linked files via the Links panel.
Thanks for the quick response...will have to try your ideas...thanks again.
and what if I do want to clean up a few things and put some of the linked
files into different folders -- but all within the same folder? Any way to
keep them linked?
If you change the path to a linked file, you break the link.
It sounds like you want to reorganize the linked files by adding a new folder level. Obviously, that's best done before links are established. What you're asking would require some re-linking afterward. Later versions of InDesign will find additional previously-linked files in a folder when an initial re-link is performed, so you wouldn't have to re-link each file; just one per new folder.
Okay - thanks for all your help. I was afraid that there was no easy way to
re-organize my files...
But thanks so much.