The most obvious risk in polyamory--other than risk of STDs, which seem in my experience to be much more rigorously policed by polyamorists than by most others--is straightforward social rejection. People think you're nuts and/or immoral, and they respond in any number of ways. Particularly if these people are your parents or employers, this can be a significant consideration. And there's always the chance that some righteous individual will decide you must be punished and attack you or something. I am not aware of any incidences of this, though; honestly, there isn't enough general awareness of polyamory to give rise to any serious poly-bashing. Maybe that will develop in the future. Or maybe we'll get lucky and skip that part.
Then there are some legal considerations. Companies conventionally give their employees health insurance as well as money in exchange for their services; it's weird that we arrange things that way, but it's what seems to have evolved, and that fact plays a rather aggressive role in defining family in our culture. Your employer will buy you some insurance, and some more for your family members--but, for that fairly understandable reason, they get more than a little snippy about who is or is not to be considered a family member. I can't name a solution to this one. Obviously I can't ask anyone to insure all people. But there is something arbitrary about a company which would willingly foot the bills for four accident-prone children, but not for two spouses.
Then there's power of attorney, another clear problem. A poly lawyer once told me that courts are generally willing to try and work with you, if you've made (and, importantly, recorded) any old sort of agreement--handfastings or whatever, state-sponsored or not. And I do believe it's possible to endow someone not legally related to you with power of attorney in the event of some emergency. I don't know if that legal procedure will permit such an endowment of more than one, equally--for understandable reasons. Anyway, the fear is of the parents or other relatives, chiefly, seizing power over a lover no matter how estranged they might previously have been.
The real fear of polyamorists, though, towering over these others, is loss of their children. And this is the thing that puts teeth in that basic social unacceptance. It doesn't take much to cast suspicion of child abuse; almost anyone can make the allegation, based on whatever they may claim to have seen. And it only takes one judge to issue a court order remanding your kids to state custody. A lot of people believe that anyone not monogamous is already perverted; they will readily suppose that one perversion leads to another. It's already happened once. More is probably to come. Or not. Maybe people will start to hear about poly and they'll shrug and not worry about it and feel no need to oppose it. Or not. There's no saying.