From Brittanica.com
As early as the 3rd or 4th century ad, in rites held in the cloisters of churches, parishioners may have placed their ever-present club, or Kegel (the implement most Germans carried for sport and, certainly, self-protection), at one end of a runway resembling a modern bowling lane. The Kegel was said to represent the Heide (“heathen”). A stone was rolled at the Heide, and those successfully toppling it were believed to have cleansed themselves of sin.
If this is true, was it a ritual unique to Germans? Or did other cultures practice it as well? How long did the religious aspect of it last, and how did it turn into the modern score keeping competitive sport we know of today?
I was intrigued by your question, and decided to do a little research (it's a little slow at work today).
Turns out that the notion of "bowling" being present in Germany in the 300s AD is traced back to the work of William Pehle, a German bowling enthusiast in the 19th century. He wrote a work called "bowling" which is cited in this 1949 article from the Ottawa Citizen newspaper. Also, he is cited in the full text of 1958's Modern Bowling Techniques (p. 72-73).
I am unable to find out where Pehle got his information, or to verify the truth of his statement. All I can tell you is that this has been the story since the 19th century, and William Pehle is the source of it. As someone who studies history, I am uncomfortable with calling this true, as it comes from a secondary (possibly amateur) scholar whose work I cannot find to evaluate, but I also haven't found anyone denying the story either.