Having gone through weeks of my Genie HR44 dropping its wireless network connection and having to go through internet setup several times to reestablish the link, I've made one change that has made all the difference.
While setting up a network connection between my router and the Genie, I had been pushing the "WPS" button on the router and allowing the DVR and router to link automatically.
However, when I used the manual procedure (entering the password key for the router using the remote), the connection was made successfully and continued uninterrupted since.
I've seen suggestions for switching to ethernet and even powerline network adapters but I didn't see any mention of this possible solution. Just wanted to share that for anyone in a similar situation.