Payment of principal only?

Yes, you have the right to request that it be paid to the principal only.

Thanks, JCEMT! Is there a particular law or code that I can reference when I make that request to them?

Can't quite remember but it may have been truth in lending. You can check FDIC.gov to check on it.

How the payments will be applied is spelled out in your card holder agreement.

Guest there is FDIC statute which allows you to specify. I once worked for Bank of America as a personal banker and I know what I'm talking about.

Just paying the principal will not bring your account up to date it will help out some with the interest though in the longer run.

First, with credit cards it doesn't matter. Credit cards are balance-forward accounts, and the interest is compounded every month, so your payment is not applied to any interest versus principal. Payments are just credited toward the "statement" balance and then carried-forward every month.
What you are asking for is basically for them not to charge and compound any interest when they carry the balance forward. You can't exactly tell them to stop doing that -- the only way you can get them to stop compounding interest is by paying it off or possibly by them charging it off.
JCEMT, I have to disagree with your statement.
Installment contracts and mortgages are generally the only type of loan where you'll see payments applied as a combination of principal versus interest. Specifically, you'll only see that in simple interest-bearing contracts (as opposed to the Precompute/78ths method).
On simple interest notes, a debtor does not have the right to demand payment be applied to principal. Payments will always pay for the interest first, before it touches the principal.
Illinois Laws are very clear on this:
and the remainder of the payment applied to the unpaid principal balance; provided however, that if the amount of the payment is insufficient to pay the accumulated interest, the unpaid interest continues to accumulate to be paid from the proceeds of subsequent payments and is not added to the principal balance. ??
Also from the Administrative Code:
interest is paid to date or is agreed to by the licensee, except a payment may be credited to principal if the amount of the payment is not sufficient to pay the interest due for one day. ??
I'm pretty sure that Regulation Z (Federal) mimics the above state laws. Off-hand I know that Reg Z allows simple interest method of computation. If you give me a moment I'll go look it up.

Must be just my state then, or IL has a different law. I assumed Reg Z (are you sure it's Z? My memory is a bit fuzzy on FDIC laws since I haven't been in banking for over 6 years) was the same as my state statutes.

Thanks everyone! So does that mean that interest compounds on interest and late fees? Somehow I thought that wasn't the case. But if whatever payments I make are applied only to the balance due and not to interest or principal in particular, then I don't see how interest could be computed on anything other than the full balance.

That's generally how it works.

Again, the reason I despise credit card companies. They are "legal" loan sharks. People need to quit the plastic and stick to paper.And I mean cash, Mo's and Checks (good checks).
It'll be a rainy day in hell before I get a credit card. (I use a debit card, works like a credit card, but no fees or interest, as long as I do not overdraw my bank account)

jcemt..you are very young looking indeed from your gallery pic to not have been involved in personal banking for past six years- are you indulging in some sort of youth serum :D