you could make up food allergies, but that gets tricky...like "you can eat french fried but not mashed potatoes?" so i wouldn't recomend the allergy approach. Also, try not to use the phrase "picky eater". It makes people think of spoiled toddlers. They need to understand that your preferences and reactions to food are beyond your control.
I was in italy (i'm american) for 6 weeks this summer with a small group of other american students. We all ate the same food together so i had to explain myself pretty quickly. I told people that i'm extra-sensitive, that i can hear and smell things that other people can't or that don't bother them. They said "wow, you're like a super-hero!" I told them, similarly, because of sensory issues i can only eat a few certain foods. People were usually very curious and asked questions about what i can eat, and like is it taste, or texture? I told them its taste, texture, temperature, and smell. They usually sympathised with me and throughout the trip were continually curious, like "can you eat this? how about this?"
So my advice is to just give them a simple but honest explanation of your sensory problems with food, without going into a diagnosis or other problems you may have. Make sure they understand that you are not just being picky, and that it isn't their cooking. I made sure to mention that if i eat something i "dont like" i may actually throw up, so its not a choice.
I also recomend that you let the family know about this issue beforehand, so that you don't have to explain it at your first meal. Write them an email or something before you get there, with a list of foods you do like, and offer to cook your own meals sometimes so they don't have to go out of their way to accomodate you.
Good luck!