I needed to grep a directory, and sub directories, and not displaying the filenames in the output. This is when the man page came to the rescue.
man grep | grep filename
-H, --with-filename -h, --no-filename
There we go, -h, --no-filename, is what I needed.
Example with output to a file:
grep -r "searching-for-this" . --no-filename > /tmp/test.txt