My source grew up on a farm in northeast Nebraska and recalls learning this indicator when he was 7 or 8 years old. His grandmother owned three dogs during his childhood, and one day he saw them all eating grass at the same time. He found this odd, so he asked his grandmother if she forgot to feed the dogs. She hadn’t, and explained to him that when dogs eat grass, it’s an indication that it will rain soon. Sure enough, it rained later that day. Afterwards, most of the time he saw the dogs eating grass, rain quickly followed.
It is not out of the ordinary for a dog to eat grass, and it is actually typical if a dog has an upset stomach. But then again, a coming rain is not likely to make a dog sick. My informant suggested that there might be an atmospheric change that occurs before a rainstorm that might make dogs believe they have a symptom of an upset stomach, so then they would decide to eat grass. There is no proof to support this explanation, but it makes sense to my informant considering the likelihood of rain after he saw his dogs eating grass.
However, there were several times that he would see the dogs eating grass and it wouldn’t rain. In these cases, either the dogs were sick or it was a dry season. This supports another folk superstition that his grandmother once told my informant. She would say, “In a dry spell, all signs fail.” My informant’s grandmother knew many folk superstitions, and she would tell them to the family when appropriate. No one else in the family desired to memorize them all as she had done, but they would remember the ones that she had told them over and over, and they shared those between each other. These superstitions were likely shared in the same way by many other families. This particular superstition is likely to be shared mostly by farmers because their occupation and livelihood is dependent on weather patterns, so if there is any way farmers can make use of a weather indicator, they certainly will.
Annotation: This particular folk superstition can be found in John Frederick Doering’s article: “Some Western Ontario Folk Beliefs and Practices” in The Journal of American Folklore, Vol. 51, No. 199 (Jan. – Mar., 1938), pp. 61