Good news for ladies who like the woods—your period is (probably) not something that attracts (most) bears.
There are not a lot of studies addressing this particular topic, but a National Park Service paper published this year took a look at all of them and put the scattered pieces information together into a single puzzle. It's probably not a complete picture, but it's certainly better than hearsay and random, sexist stories you heard from your grandpa's drinking buddy. More importantly, even when there is a documented risk between menstrual blood and bears, that shouldn't be construed as a reason to keep women out of the wilderness. After all, bears are attracted to food, and we don't tell people they shouldn't eat while backpacking. Instead, we have practices that reduce risk. Same thing applies here.
Here's what we learn:
1) You can menstruate freely and without fear in the contiguous 48 United States. Grizzlies, and particularly black bears, don't seem to be interested in what's happening in your pants. Evaluating hundreds of grizzly attacks found no correlation between menstruation and risk of attack. In the case of black bears, this has actually been tested experimentally, with researchers leaving used tampons from various stages of menstruation out in the wilderness and watching how the bears respond. (Science!) The bears completely ignored the tampons.
2) Yellowstone data suggests food is a much bigger risk than menstruation. Analysis of bear attack data from Yellowstone National Park doesn't even consider attacks that happened before 1980. Why? Because that was before stringent rules on in-park food storage and bear feeding. The vast majority of pre-1980 attacks are already known to be related to bears seeking out human food. Meanwhile, between 1980 and 2011 only 9 women have been injured by bears in Yellowstone. Of those, six were incidents where women and bears ran into each other unexpectedly on hiking trails. In the other three incidents, which didn't rely on the element of surprise and are, thus, more likely to have attraction factors involved none of the women were menstruating at the time of the attack.
3) Polar bears are a whole 'nother story. Two different polar bear studies, one in captivity and one in the wild, have shown that those bears are attracted to human menstrual blood—even more than plain old human blood that wasn't related to menstruation. They are also attracted to the scent of seals and (again) human food.
Big picture: Food still seems to be a bigger issue in bear attacks than menstruating ladies. And, as with food, the Park Service has guidelines that you can follow for how to best deal with tampons while in the wilderness.
Read the full National Parks Service report, including the safety guidelines for women on their periods