There is a popular saying that turkeys will drown if they look up in a rain storm. Is this actually true?

Not according to the sources below. But apparently, they can be scared to death.

"RUMOR—Turkeys are so dumb that they will look up when it rains and drown. This popular belief is simply not true. Even the domestic turkey does not look up at the rain and drown. This rumor most likely started from farmers who had domestic turkeys that died during a rainstorm. But instead of drowning, the birds were probably scared by the lightning, panicked and congregated en masse into one corner of the pen, suffocating the unfortunate birds in the center."
https://web.archive.org/web/20071110233633/http://nwtf.org/tips_adventures/adventures.php?id=250

While turkeys don't literally drown in the rain, there is a grain of truth to the idea that they do. Until they're about eight or nine weeks old, baby turkeys are covered with down rather than feathers and consequently are quite vulnerable to the effects of weather. In the wild, the baby turkeys' mommas hustle them out of harm's way when a storm blows up, but domesticated turkey mothers, having been rendered spineless by generations of welfare, are apparently too indolent to do so. (In fairness, modern poultry raising being what it is, Mom may no longer even be on the scene.) As a result, the chicks get cold and wet and often die of exposure when it rains.

https://web.archive.org/web/20130529063756/http://www.discovergames.com/butterball.html

No, it's not; in fact, I believe just the opposite is true.
(I could be mistaken on this, though. I started with my own experience and observations and then went looking for verification.)

"Contrary to urban legend, commercially raised turkeys are NOT so dumb that they will look up at rain until it fills their nostrils and they drown."
http://www.hiltonpond.org/thisweek031115.html

But, I'm more familiar with chickens. Many and many a time I've seen chickens point their beaks skyward in the rain, and we figured out--perhaps correctly, perhaps not--that they were preventing themselves from "drowning," or, more accurately, aspirating rainwater.

A chicken's nares are on top of his beak, and rain would naturally fall into them, or at least that seems apparent to me. As far as I know they cannot be closed.

Raising turkeys is similar to raising chickens, but more troublesome.