Answer:
I heard that your dog was put down last week.
Explanation:
because when you say " I heard that you had to dispose of your dog last week" that doesn't sound like you care, same with " I heard that you euthanized your dog last week" whereas, "I heard that your dog was put down last week" sounds like you care. it sounds a little more like you care than " I heard that you put your dog to sleep last week" and the other two.