Before this, I never noticed there wasn't an option to assign a silent ringtone (or to not assign any ringtone). Kind of an odd omission on Apple's part, in my opinion.
Either way, this (link below) appears to be the only option, from what I could find:
This is such a useful feature that Im hoping Apple includes a None option for ringtones in a new version of iOS sometime down the road, but until then this blank ringtone approach works just fine.