I know that many people and churches prefer using translations other than the KJV. Why? What are the common criticisms against it?
The usual criticism is readability. The KJV was originally translated in the 1600's, and even though it's been updated several times since then, there are still words that have different meanings now than they did in the English of the time. The literary style is clearly less modern than more recent translations.
Another criticism is based on the translation sources. The New Testament portion of the KJV is based on the Textus Receptus (Received Text), which has been subject to some criticism. This has resulted in some minor textual differences that can be traced back to the different source manuscripts.
I hesitate to answer more completely because the choice of a Bible version is personal, and I want to avoid stating that any version/translation is superior to others. There are those that feel very strongly about the superiority of their own version (KJV-only and those strongly opposed.) The textual differences mentioned above tend to fuel such arguments, which in my opinion, tend to be counter-productive.
Another criticism is the distinction between a translation of the original texts, and a paraphrase. The various Bible translations fall somewhere on a scale between word by word equivolence and thought for thought equivalence or paraphrase. The KJV is closer to the word-for-word end than many modern counterparts.
The KJV-only groups tend to prefer a direct translation of the original because it is less subject to be influenced by the views of the translator. They argue that an interpreted version is suspect, because the result is influenced by the translator's understanding of the meaning of the text. On the other side, the argument goes that the "paraphrase" versions are written so that the meaning is clearly understood, and it's therefore "easier to understand" than the "archaic" KJV.
The biggest problem with the KJV is that no one is fluent in its language any more. It simply is not written in an English anyone speaks today.
When you read the KJV the problem isn't words you don't know - you can go look them up in a dictionary like you do any other words you haven't read before.
The problem isn't complicated sentences or strange word orders - you can spend the extra mental energy to parse it even if you're not reading smoothly.
The problem is that words change what they mean over time. You can learn what changes have occurred, and sometimes you'll realise that the words don't mean what they think you do because they don't make any sense in context. But often both meanings fit the context and unless someone has told you that a word used to mean something else you'll just automatically read it with the modern meaning. So the biggest problem of the KJV is the one you don't realise you have!
An example of this is the word 'conversation', which is found in Philippians 1:27 in the KJV. The modern meaning of 'conversation' fits the context perfectly, but it's not the right meaning. Every modern translation, including the NKJV, translates it as 'behaviour' or 'conduct' or something like that. Unless you already knew that 'conversation' used to mean something else you would never think that it did from reading it. Later in Philippians 3:20 the word 'conversation' is used to mean 'citizenship'. I don't know if it used to have two meanings or was just translated poorly, but again it's the same problem. That verse doesn't make quite as much sense, so you might guess that something weird was going on, but you might not.
For five more examples, try this quiz: Do you speak KJV?
If you're willing to check every word you read in the KJV against an Early Modern English dictionary, then go ahead and read it. But if you're aren't, I guarantee that there will be times when you understand it incorrectly without even realising. Please, read a Bible translation in a language you actually know!