This is a much more complicated question than it seems at first sight. There are at least 3 things "Troy" may refer to:
Depending on which of these Troys we're looking for, we'll need to apply very different standards of accuracy. The main problem is the fact that the famous works of Homer consist of a glamorized account of the life and warfare of Early Archaic Greece (c.800-650 BC) rather than a historical account of the wars of Late Mycenaean Greece (c. 1300-1100 BC). So, should we judge the accuracy of the movie Troy by its representation of the 12th century BC, the time when most Greeks believed the Trojan War took place? Or should we judge its accuracy by its representation of what we actually find in the Iliad and Odyssey?
The answers so far ITT are confused on this issue. Some of the inaccuracies listed here test the movie by the content of the epics (or the wider epic cycle), while others test it by what we know of the technology and culture of 12th century BC Greece. They skip over the fundamental question whether we want to judge Troy as an account of a historical war, or an adaptation of an ancient epic poem.
If we're going with the former, inaccuracies are easy to find. Most of the equipment and objects seen in the movie are wildly out of place for the 12th century BC. Many warriors use iron weapons, which were unknown to the Greeks at the time. No one is seen bearing typical Mycenaean types of helmets or shields. As noted by /u/Mithras_Stoneborn, coins were still some 600 years away from being invented. While there are Mycenaean depictions of armed men on horseback, it seems very unlikely that they knew of organised cavalry of the type seen in Troy. The list goes on.
If we're going with the latter, however, things get more interesting. A lot of the armour and weapons seen in the movie fit much more closely with what is described in the epics; some of it is clearly inspired by archaeological finds dating to the late 8th century BC. Iron was the standard metal used for weapons in this period, and depictions of mounted infantry are widely known. The movie shows chariots being used as the poems describe. Coins are still out of place, but the construction of large funeral pyres for dead heroes is very much the norm in Homer's works.
More importantly, the movie portrays the nature of combat very much as it appears in the Iliad. Large masses of men advance against each other, but the action scenes focus on the deeds of heroes with almost supernatural strength. The tide of battle depends on the presence and actions of these heroes. While there is a clear divide between the heroes and the regular warriors, there is much more social and cultural equality between the heroes of the two sides; they make agreements among each other, share ideals and habits, and let honour override expedience in their dealings with one another.
In the epics, of course, the gods play an important part, and this was cut entirely from the movie. The individual plotlines of a lot of characters are also fundamentally changed. On the other hand, each of the human characters we see on screen stays remarkably close to the way they are portrayed by Homer (with only Agamemnon and Menelaos having apparently taken a degree in evil). To the glee of those who have read the poems, the Iliad is actually cited verbatim several times.
Whether this makes it "accurate" is really up to you. But it's important to realise that its supposed accuracy is a very complicated matter, and it would be very unfair to simply declare it inaccurate because it's missing or adding one thing or another. It's worth asking yourself exactly how any director could make an "accurate" depiction of the Trojan War.