Can grep show only words that match search pattern?

Is there a way to make grep output “words” from files that match the search expression?

If I want to find all the instances of, say, “th” in a number of files, I can do:

grep "th" *

but the output will be something like (bold is by me);

some-text-file : the cat sat on the mat  
some-other-text-file : the quick brown fox  
yet-another-text-file : i hope this explains it thoroughly 

What I want it to output, using the same search, is:

the
the
the
this
thoroughly

Is this possible using grep? Or using another combination of tools?

Try grep -o:

grep -oh "\w*th\w*" *

Edit: matching from Phil’s comment.

From the docs:

-h, --no-filename
    Suppress the prefixing of file names on output. This is the default
    when there is only  one  file  (or only standard input) to search.
-o, --only-matching
    Print  only  the matched (non-empty) parts of a matching line,
    with each such part on a separate output line.