Have you just been going out there, cutting their grass while assuming they still want it done by someone else?
In my first year, I took over 18 of my best friend's customers and since birds of a feather flock, HIS customers were somewhat like him and since he is my friend, they were also somewhat like me. Still I lost 2 right off the bat: One plain did not want to go with me, the other we could not see eye-to-eye. Expect this in your case as well. Also my best friend and I drove to each customer's house and he introduced me to them individually and walked me through the whatever-you-want-to-call-it procedure one-on-one. This helped build trust.
So advice-wise, I can tell you how to get down and dirty with the collections affair, but I really think a softer approach would work best:
You want to STOP all service of those who have not paid at this time until you get this resolved BUT do not assume the service is over and do not have an attitude, this is only for you to control your debt management. Plan on not getting paid for some but again, no assumptions just chill.
A humble, be nice and be yourself, non-arrogant approach works best -
I think you can chill with the guys who have been paying but as for the rest, I would recommend calling ALL of them. I would ALSO call anyone whom you have NOT serviced (if any) and doing the same.
So then say Hi and tell them your name and Explains to them what is going on / what has happened. Do this for ALL of them and leave a message if they don't answer:
Hi, I am FIRSTNAME who has taken over the lawncare aspects of (name-of-business or the guys) <<optional: and Have been cutting your grass>> ... I am new at this and it crossed my mind some folk may or may not want me to service their yard, so I am calling to see if you would be interested in my continued services. So if you would be so kind as to please call me, my phone number is 888-555-1212 and let me know, I would greatly appreciate this. Thank you, and have a nice day.
That should get you something, do that and then get back here to let us know the results.
Good luck.