I like to take the perspective that all press is good press. So at one level, I just let it slide. If it negatively affects certain relationshps, then that's on the other person, not on me.
But, if it is getting to the point of being very serious -- such that authorities may be or need to be involved -- then you need to ask for a mediation before it gets to that point.
For a mediation, you will need two things: evidence and confidence.
Evidence is the tougher one to get, but the best evidence is to simply make a note on your calendar of the rumor that your heard, when you heard it, what it was (in as much detail as possible), and from whom you heard it. If possible, ask them from whom they heard it. If possible, ask that person where they heard it. If any of these people heard it via email, then see if someone is willing to forward it to you. Keep an electronic and a hard copy.
You are creating a file here, because rumors can step into the realm of legal harassment, and you can be protected from that (and the other person punished for it). Evidence is *key* to this process, though.
I have had several stalkers over time, and so I have extensive evidence of their practices, and currently, there is an issue brewing well outside of our business. We have recently won two court cases, but then had to take it to the next level to get paid (eg, court collections, which creates a certain level of public embarrassment). Since this happened (and we got paid), there have been several "odd" happenings around our business. There was a complaint into city council about our sign, along with the accusation that we didn't have the right license for our sign. Of course, this is false. I informed them of the situation -- as I suspect that these are the folks doing this. The same happened with our debit card machine process -- someone called in to "cancel" our account, and when the company called to 'verify' the cancellation, of course we knew nothing. And since only two of us can do that (as directors) and neither of us did, we also informed them of the situation with these people. The post office called to verify that we were shutting down and have no forwarding address -- of course this isn't true either.
But I do have notations about each of these calls in my calendar, and I asked for evidence. I asked the city council to send a copy of the complaint; I asked the debit card company to send a copy of the order (including our cancellation); I asked the post office to forward the document they received.
Obviously, something is "up" and this does qualify as legal harassment. I'll keep notes until it either A. dies down, or B. the person doing it reveals herself and I have the evidence to file a police report and get the court involved. My lawyer also has a copy of everything relevant in these files.
So, evidence, evidence, evidence.

If it's going to go that far.