The Oxford English Dictionary definition for racism is as follows:

The belief that all members of each race possess characteristics, abilities, or qualities specific to that race, especially so as to distinguish it as inferior or superior to another race or races

Racism is far from being a modern concept; one race holding stereotypes about another is evident even as early as Ancient Greece. For instance, there were many different kinds of Greeks; Spartans and Athenians were just two "types." Although there were significant differences between the Spartans and the Athenians - and they butted heads so many times over history - they both considered themselves Greek. What united the Greeks? A common language, commonalities within culture, and a common enemy.

The Greeks considered the Persians to be barbarians (negative connotations); they even called all foreigners "bar-bar." The Persians spoke a different language and they wore different clothing. The Greeks drank wine which they considered the drink of the gods; they considered beer to be the drink of barbarians. They had this idea that they fought for one another, for the good of Greece, while Persians fought so that they wouldn't get punished by the king; this made them see Persians as particularly weak and by some accounts even effeminate.

These differences really came to light as Panhellenism came about; they became more aware of what it meant to be Greek after coming into contact with so many other nations. So even though Athenians and Spartans - for instance - were quite different, they knew they had more in common with each other than with Persians. Panhellenism almost literally means "all Greece" so it's evident that Greek identity and this concept of "otherness" - in other words non-Greeks - meant a lot to them.

To answer your question, no, racism by the above definition isn't a modern concept. The Greeks stereotyped others, had a very solid understanding of "us" vs. "them", what made other cultures inferior to theirs (drinking beer for instance), and really did believe in their own superiority. Then again, every ancient culture I can think of did this in some way and to some extent. We also have to be careful in calling this racism because society then was very different from society today.

Source: "The Greeks: History, Culture, and Society" by Ian Morris