This is very easy to do in a terminal.
First, create a new folder for the files to go into.
Then open Applications -> Utilities -> Terminal.
Then type in the following, but replace the underlined parts:
find source -type f -exec cp -p {} destination \;
Replace "source" with the path to the folder containing the files.
Replace "destination" with the path to the new folder.
You can drag and drop the folder into the terminal window, and it will type out the path for you.
After the command completes, you will be returned to the prompt, and the new folder will have a copy of the files. Be aware that if you have more than one file with the same file name, only one of them will be in there. If you're satisfied with the results, you can trash the old folder.