This is quite a bit after Poe's time, but the researcher famous in the 1960s for the notion that plants communicate was named Cleve Backster. The Wikipedia article on Backster refers to the theory of "primary perception", which term may lead to more information.
I found a few terms that seem not quite to fit, but are used in the discussion. The most likely is hylopathism. But this term, like several others (holism, animism, panpsychism) is much broader, including a much wider range of the material world, and not only plants. I haven't found any one-word term for the belief that only plants and animals are sentient. Most discussions seem to use the phrase plant sentience.
This is probably more along the lines of Glenn's suggestions, and not specific to plants, but I've encountered the concept of the consciousness continuum in more than few places. Here's one reference from another forum I hang out at: http://www.physicsforums.com/showthread.php?t=13077
The basic idea is that the sentience humans experience is neither the highest nor the lowest level, but merely one point in a spectrum. The claim is that, for sentience, all that's needed is an exchange of information. More information exchange = higher sentience.
Not sure I'm willing to buy into that, but it does raise the fascinating question: If some species were more sentient than us, what would that feel like (to them)? Now there's a good question for the philosophers to ponder.