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?