Heirloom plants are those that were in cultivation at least about 3-5 (human) generations ago, but aren't at present due to changes in agricultural trends, and the present day reliance on seeds with certain desired qualities (at least, in large scale cultivation). As such, these heirloom plants are well adjusted to the local area where they once were in cultivation and seldom do well anywhere else.
The most common mistake that people make is to try and grow just any heirloom variety (for the sake of growing a heirloom), without consideration as to where it used to grow. In other words, you can't expect to grow a heirloom Mexican pepper plant in, say, northern Europe. Just ain't gonna happen.
Personally, I've only grown heirloom peppers and I've found them to be very easy to grow, since my climate is fairly similar to its original environment. The one I'm growing right now is the pepper in this question (Pasilla bajio). It has a very mild and flavourful taste, and works great in salads for a little kick (yes, you can eat it just like that!).
I guess no matter which kind of plant you want to grow, the important thing to make sure is that they suit your locale.