Understandable. Each week, someone is responsible for the maintenance. We have a maintenance schedule with each client, and if there is any on site maintenance needed that week, we send a tech out to perform it. Freeing everyone else up for the other stuff. Right now, all sites are assigned to me. But as we grow, and have more contracts and a lot less break/fix with some of these clients, I will assign tech's to them. I'm a firm believer that you still need foot soldiers in the trenches who meet with clients, this goes back to my how I'm my clients therapist.
This is why I believe in and how I plan on assigning sites to people:
1) Assign a person who will know the hell out of that site/network/etc. They do not need to be the only person who maintains that site, but they are the site expert. They know everything that client has, how it is supposed to run, etc. But this employee, he is responsible for the maintenance, and he is responsible for relaying any special information/assistance to the other techs. Have a Sys or Net Admin, this is their job. It also removes some stress off of you, and places you more in management mode and less in field mode. Don't worry, you can still go out and turn wrenches.
2) Make that person meet with the client, and build a relationship with them as well as with you. Your techs represent you, your values, and your company. Use them as an extension of you.
3) If this person quits, gets fired, or is no longer capable of working for whatever reason, promote within, and send someone there immediately to take their spot. Build and keep that relationship, and always have a site expert.
You don't need a lot of people to accomplish this, you said you got 5 guys working for you. Take 1 (or 2) and assign them sites. Tell them which sites they are responsible for, that they are responsible for the maintenance (not the break/fix), and that they meet with the client once a week to see if there are any issues that the tech may not be aware of. On my end, I see everything is working fine, but I had no idea that Joey in accounting was having a problem loading his accounting software, or printing and never got around to calling us about it. Tell them to learn the the network and all current settings, etc, and put it in a book specifically for that site (We have a PDF book that covers the site, and then a physical hard copy in a .5" binder). This book is your property, but the tech is responsible for maintaining it, this is the in case someone new has to do anything with configurations, or how things work.
Try it, and let me know.