The date September 15, 1774 was carved beneath the cabinet.

I need to know what words modify the
predicate. I think beneath modifys the
predicate but is there another one?

The word "beneath" in this sentence is a preposition. Therefore it has its own phrase, a prepositional phrase: "beneath the cabinet." The entire phrase modifies the verb. The entire phrase is part of the predicate.

verb = was carved
complete predicate = was carved beneath the cabinet

Please repost if you have more questions.