Most of them are bad reasons - for example, one very popular criticism is the claim that KJV invented the usage of the word "unicorn" in the Bible, and it's actually a wild ox. But it was traditionally translated that way from the time of the Latin Vulgate, and nobody knows for sure what a re'em is. [Ed - it turns out the Septuagint also uses the word μονοκερως, so it goes back even further.) So while it seems reasonable to personally disagree with the choice of word, it's going too far to declare it to be wrong when a) nobody knows what the referent is for sure b) the two pre-existing "canonical" translations have used your equivalent translation for ~1800 years.
You can find people explaining why not to use any translation; people are, uh, religious about their chosen translation and the KJV is the old guy you've got to tear down. But any mainstream translation is probably OK. You certainly miss out on a lot of poetry and historical citations if you skip out on the KJV, and there's very few actual problems with it. Probably the fairest criticism is the general decline in reading ability means that many people find the KJV hard to read, and the NKJV in that regard is only a minor improvement for modern audiences.