I didn't want to say anything until I was back home, but bringing everything in my carry on bag in both directions, worked. So, I'm now home successfully with 8 vials of DNA sample obtained from relatives in a foreign country.
I actually phoned FTDNA about this issue before leaving and they assured me that it wouldn't be a problem and very kindly emailed me a simple PDF letter stating that the samples are indeed just that:
The letter is on FTDNA letterhead, dated and signed, with phone numbers, etc. I would suggest that anyone wanting to do the same just call FTDNA and ask for one at the time of their travel. They emailed it to me just minutes after the phone call.
Being such a simple letter, I typed it into Google Translate and attached a Spanish translation to it as I was going through customs in South America. So, I kept a copy of the FTNDA letter and the Spanish translation with each one of the kits, but I never needed to show it to anyone.
I'll feel good about doing the same thing in the future, which I will probably have to do sooner rather than later...
Update Dec 19 2016: I went to a holiday party today and happened to meet a former TSA Airport employee. He had all kinds of interesting stories, but of course I asked him about this issue. He said in general, it should be fine. It's a big and significant help to have an official letter from FTDNA on hand, just in case. My Google Translate Spanish letter is a thoughtful gesture, but it isn't necessary as there should always be someone on hand to interpret such documents in any language. If somehow a TSA or customs official does have an issue with the vials, or wants to open them, etc., you should immediately ask for a supervisor and explain the situation to them.
However, beyond all of that, the former TSA employee said that if you truly don't want any chance of a hassle, you're better off following @lkessler's original advice and shipping the DNA back home. Just sending it through in your checked luggage is another option, but one I don't think I would ever do.