That little nugget of wisdom comes from the movie Anchorman, but that's not its source. It's often repeated, and likely stems from a bear attack in 1967 that left two women dead and many people speculating that the bears were attracted by menstrual odors. Is there any truth to that claim?
Brick Tamland [opposing women in the newsroom]: I read somewhere their periods attract bears. Bears can smell the menstruation.
Brian Fantana: Well, that's just great. You hear that, Ed? Bears. Now you're putting the whole station in jeopardy.
Research has been done on the subject. In 1991, a study was done showing that black bears are uninterested in human menstrual odors. This was done with a series of experiments involving either used tampons or menstruating women in close proximity to the bears. The results showed no preference of the bears for tampons over garbage or feed. There was no preference for used tampons over unused tampons or those soaked with non-menstrual human blood, and no difference in interest with the lower torso of menstruating versus non-menstruating women. In all cases menstrual odors were virtually ignored. For grizzly bears, no scientific studies have been done, but a retrospective analysis of grizzly bear attacks on humans showed that there was no evidence linking menstruation to the attacks.
One place where an attraction to human menstruation may be true is in the polar bear population. In a 1983 study, polar bears both in captivity and in the wild showed a preference for food scent samples or used tampons over unused tampons or non-menstrual blood suggesting that polar bears are attracted to menstrual odors. A brief summary of these studies has been published by the bear management office of Yellowstone National park and can be read here (pdf).
Overall, the evidence indicates that unless you're heading pretty far north you don't have to worry about menstruation attracting bears. The Museum of Menstruation and Women's Health backs up that claim. What if you're into SCUBA-diving instead of hiking? Sharks are notorious for their attraction to blood in the water. The Museum of Menstruation has something to say about that as well: "many women fear that this discharge may attract sharks. The truth is that women divers are attacked by sharks less often than men are." However, there could be a number of reasons for this difference, and whether sharks are attracted to menstrual bleeding remains inconclusive as no controlled studies have been done. There are arguments to be made for both sides, as discussed here, but the consensus among divers, at least, is not to worry about it.
So bears are indifferent to menstruation and we're unsure about sharks, but what about people? There are behavioural cues, but no solid data regarding pheromones. Other phases of the menstrual cycle are better studied, including the time of maximum fertility just before ovulation. Many studies have shown that women at this point in their cycle, 'hidden-estrus', appear more attractive to men by a number of different metrics. (Some studies were mixed-sex and women found them more attractive as well!) This can be accompanied by other behaviours, such as men being more attentive or protective of their partners. The effects of 'hidden-estrus' signals on male behaviour is highlighted in this recent paper in Evolution and Human Behavior. In this study, 18 strippers reported their tips and menstrual cycle over a period of 60 days. It was found that among normally cycling strippers, peak tip earnings occurred during 'estrus' compared to the luteal phase which, in turn, were higher than during menstruation. This difference disappeared among strippers taking birth control pills. A deeper discussion of the paper can be found over at Pure Pedantry. For people who may not be familiar about the inner workings of a strip club (or, as the authors put it, "Because academics may be unfamiliar with the gentlemen's club subculture...") the paper even includes some background on the US stripping industry that alone is worth the read. Despite it's inherent humour value, this study could be a jumping off point for some interesting discussions about human evolution and behaviour. Have humans actually lost the 'estrus' phase, or is it just better hidden than in animals? What are the evolutionary advantages? Or it could just be an excuse to do more science in a strip club.