Hi bworthey.
I'm not KJV-only, and I regard myself as somewhat proficient concerning the hundreds of English Bible translations available today, including the great many of them within the KJV family of English Bible translations.
There's indeed some variety within the KJV-only camp, and you're sure to get various responses and defenses reflecting this variety. That is, various KJV-onlyists hold that position for a variety of reasons, and not all of them have to do with either accuracy or concerns regarding the original writings (or, "autographs") of the Scriptures, of which none are available today, but only copies of copies. Indeed, one of the most central of rationale behind the KJV is a preservation of God's revelation beyond the original autographs and languages of the Bible into modern English. This cuts to the heart of the question, "what makes a translation accurate?"
Apologetics regarding English Bible translations are usually made along the following criteria:
1) What are the original manuscripts? What were the Scriptures originally like? We don't have the originals. God had His purposes for not having them preserved to our current viewing. Without the tablets of Moses or actual written letters of the Apostle Paul we rely upon what preserved records we have. Most modern English translations (like the NIV, NLT, and HCSB you mention) favour the oldest manuscripts we have available, using what are called eclectic critical texts - an educated putting together of the Greek text from what manuscripts and bits and pieces of old copies that remain and have been discovered. Especially as regarding the Greek New Testament, modern translations generally make use of two primary continuing efforts toward recapturing the original text - the United Bible Societies' Greek New Testament (UBS4) and the Nestle-Aland Greek New Testament (NA27).
The KJV family of translations (like the 1611 original KJV), are somewhat unique in favouring a particular line of Bible copies used by what survived to be the majority of Christendom - what are often called the Majority Text or Byzantine text-type, because, although not as old, were preserved as the texts the majority of Christians had for their use, and which survived into most Latin texts used at the time the KJV was produced, and which underlines what is often called the Textus Receptus available to Protestants (like the Christians of King James' England) at the time medieval translations began to be made into the languages of regular people.
The KJV-onlyist thought is that since this was what the majority of Christians had for use, then it must bear the blessing of God for use by His Church. However, as logical as this thought might be, it sometimes fails to consider that just as there are some differences between all the variety of Bible manuscripts which have survived, there are also differences between Bible copies within the Majority Text family of manuscripts. There is no one ancient and preserved text, even as regarding the KJV, which, despite efforts at standardising, remains with differences even from the first printings of the KJV.
So, this first consideration has to do with what we translate from into our English language. The assembly of men working on the KJV tried to do the best with what they had access to, but they did not have access to some of the most ancient copies of the Bible, but instead favoured using what traditionally had been used by a majority of Christians, especially as reflecting the Latin Vulgate in use by Christians for over a thousand years.
2) Part of the worth of Bible translation is involved in the people doing the translating work. Were they Christians? Were they skilled and honest in their efforts? Were they influenced to tamper with the text in any way? What was the level of their scholarship and competence? The KJV is highly prized for the dedication of these Christians at a time when doing such things as translating the Bible into English often carried a steep price, as many of the early English translators in precedent to the KJV paid with their life.
3) The transmitter-specific / receptor-specific debate. That is, what matters more in translating, trying to remain as close as possible to the original language forms and expressions (even though some of their cultural idioms might be misunderstood by readers of a different culture) or trying to be considerate of the culture whose language the text is being translated into? The KJV and HCSB might be said to lean toward the former, while the NIV and NLT might be said to lean toward the latter.
4) What version or dialect or idiom of English best conveys the original Bible languages? Essentially, this principle for KJV-olyists contends that the Shakespearean or Elizabethan English of the 1611 KJV is a 'higher' English that better preserves and communicates God's Word - that 'high English' is reverently befitting the holy nature of God's Word.
5) There can be only one. This principle for KJV-onlyists contends that God has a singular presentation of His revealed written Word. On the surface this principle sounds good, as defending that there can be only one truth, but sometimes it's taken to excess, in claiming that the KJV is even better than the original, or in failing to valuably consider that the Bible is also available in languages other than English, and while it may be true that English has become quite an internationally prominent language, especially among Christians, such a claim may belittle the Christian worth of other languages and cultures.
There are a variety of other KJV-only points of emphasis, but these are some of the more important ones. Personally, unless a Bible student can sort their way through the original languages of the Bible, I find it best to make use of a variety of English translations, and many of the modern ones, like the NKJV, NASB, NRSV, ESV, and especially various study Bibles available today make ample notes of what the major differences are between varying manuscripts. I grew up on the KJV and still find it a beautiful and valuable translation, but there's no doubt that it is old and the English language has significantly changed in the past 400 years as to merit the use of more contemporary renderings as well.