How did the mighty Persian Empire influence the modern world as we know it?
The Persian Empire in its heyday was so vast and important that the King of Persia was referred to simply as the "great king." Perhaps the most important innovations the Persian Empire contributed to its neighbors and successors were essentially administrative. According to Herodotus, the Queen Atosa invented the art of letter-writing in some fashion. More reliably, the Persians had an excellent postal system, using relays, similar to the 19th century "pony express". They also were known for administrative efficiency.
As the Persians were significant users of cavalry in warfare, they also developed trousers, which are much more comfortable for riding than the flowing robes worn by the Greeks.