Check to see if you can file a harrassment or restraining order against them. Not sure what the laws are, or what exactly they have done to you in regard to threats. I know where I live if someone threatens you on three separate incidents you can go to the police and have a restraining order put on them.
You have to have real threats though. If they are really threaten to kill you on three different occassions, where I live I can simply go to the police let them know the dates/times and exactly how I was threatened. Then the police will go to them, hand them a notice with a court date and in the mean time, they are not allowed to be within so many feet or to speak with you or they go to jail. Then on court date, both parties speak before a judge with whatever proof they may have (could just be your and other parties testimonies) and then the judge decides whether to continue the order for a period of time or not.
It doesn't sound like you ever got any sort of court order or visit from the police - if that is so, then there is no law suit - you need to be notified. You are not ignoring the lawsuit - I highly doubt there isn't one - just your neighbor trying to scare you.
And to let you know - threatening a lawsuit would not be considered a threat for restraining orders - it is more a threat about harming you.