When pitched via LinkedIn, what sort of messaging actually resonated with you enough for you to read and reply?

Short and to the point. If you are selling something be upfront. Most of the time I'm not even a candidate for what you are selling, so most important, do you homework. As Samantha McKenna says, #smykm. Make some thoughtful comments on a post of mine that resonates with you. "Good post" doesn't cut it. 

If the individual was referred from someone I know, that will likely get a response because it doesn't happen often. Otherwise personalizing the message and trying to form a connection will help a lot. The vast majority of the messages I get have nothing to do with healthcare and what I and the team does. I end up ignoring almost all of them. 

If a message is personalized to both me and the company, I pay more attention to it. If it starts out "Dear R." or "R. Allen", I pretty much don't consider it. There are other companies who say they do what we do, but if someone digs in and can say something about us and why we are different, I accept and reply normally.  Sometimes some little insight, different perspective or wit causes me to investigate the person's page and I may respond if I find them interesting.

short answer - get to be known, liked and trusted by providing value to the relationships you nurture on social. then turn comments into connections, connections in conversations and only then conversations into conversions BUT don't hide what your trying to do - otherwise you connection feels the basis of the relationship your formed is false.  Adjust your banner to show your value offering. give a free something away in your featured. present lots of ways to allow people to connect with you and every 4th post tell people what you offer to allow them to open a conversation with you.  if it helps, follow me on LinkedIn as I post about this all the time for free.

1. When it was personalized and had specific pointers related to my company
2. If an agency approached, even using publicly available information and outlining areas of improvement for my function, it got my attention. 
3. When they asked for my opinion in the form of a survey or interviews etc. Of course, we all know a sales call is likely to happen but at least, they took the initiative to understand the TA's perspective on a topic.