During the 1960s and 1970s, the counterculture movement did much to impact our lives in the 21st century. Little did they know that yoga was going to be one of the results of that influence. Back in those days, most considered yoga to be New Age, "for the hippies," in short, something way outside the norm of everyday society. However, with the passing of time, it became more accepted and you can find yoga classes all over the place.
Although being an increasingly popular practice, the practicing Jew just can't jump on the bandwagon. When one decides to look at the history of yoga, one becomes aware of halachic ramifications of practicing yoga. Yoga is a practice with Hindu origins, and is about as old as Hinduism itself. As the Rebbe points out back in 1978, because of this, we have to worry about עבודה זרה (idolatry). The Rebbe's concerns were valid. You bow to your yoga instructor, as as bow to the statue of a Hindu deity while saying the phrase नमस्ते. The positions in practice retain their Sanskrit names and the traditional "Ohm" is hummed throughout the yoga session.
As long as yoga maintains its connections to Hinduism, then the Rebbe is correct in saying that yoga is verboten for Jews. However, like with a sizable amount of rulings from the Religious Right, especially the further Right you go, you come across the issue of bifurcation, which is a logical fallacy also known as the either-or fallacy or the fallacy of exhaustive hypotheses. For a Chasid, or for any ultra-Orthodox, it is the "black and white" mentality that gives us the "either/or" option. However, there is a third option, and our ancestors took this approach with the sacrificial system that had its origins in pagan nations. What did the Jews of yore do? They took out the idolatrous elements of the practice and made sure that the practice had its focus on G-d, not on anything pagan. If you don't say the names of the positions [in Sanskrit], don't chant [in Sanskrit], don't say the "ohm," or don't bow to the statue (or better yet, go to a class without a statue), then you would be fine. As one person succinctly stated, "Yoga physical exercise can be kosher as long as it remains within the context of physical fitness and stress management."
In summation, as long as yoga does not have any idolatrous elements in the practice and is solely used as exercise and stress management, then it is permissible to use yoga. However, if you in any way buy into the Hindu elements within yoga, then yoga cannot be practiced. Interestingly enough, some Jews have already made some strides in combining the physical benefits of yoga along with Judaism (see here). There certainly is a gray area with regards to this topic, like there is with many topics in Jewish law. As such, a practicing Jew needs to be aware when making a decision of whether to practice yoga because it can either be used to improve your health (Deut. 4:9, 15), which would be a mitzvah, or it can be used to commit idolatry, which is a grave sin in Judaism.