Modern Greek speakers can read aloud from ancient texts easily and can recognize many of the words. Some words will require a dictionary or extensive previous study to understand. Modern Greek also has a much less inflected grammar (similar to "church" Latin vs. classical Latin) that sometimes causes confusion for modern Greek speakers trying to parse old texts, especially in poetry where accident Greek grammar is exploited hugely for illustrative effect at the cost of intelligibility (for modern day students and possibly uneducated ancient Greeks). Greek has also undergone tons of pronunciation changes since ancient times and to the ear Plato would sound at least as bizarre to a modern Greek speaker as Chaucer would to us.

Source is my Greek classes and a modern Greek speaker who happened to be in the class. Many modern Greeks are loathe to admit to this and assume that they would be able to effortlessly communicate with the ancients, but it isn't true. A modern Greek speaker would instantly recognize that an ancient speaker was using the same language if he were paying attention but it would take a ton of effort and a little luck to get any actual communication done.