What does comin' through the rye mean?

Answer 1

Rye is a kind of grass grown for animal consumption. In the days of Burns there were no cars and most journeys were made on foot - or on horseback if you were lucky. When I sing this I am thinking of a young woman who is walking through a field of long grass. An alternative is that the rye refers to a river (the Rye Water) in Scotland. The reason Jenny's petticoats are wet is she was fording the river.

Answer 2

Rye is a grain crop.
The paths that people used from the centre of the village to their homes often ran through the fields of the surrounding farms. "Comin' through the rye" simply means when walking along just such a path -- presumably the crop grasses are tall and screen just such a meeting off from the rest of the village, so maybe the laddies were emboldened to try to steal a kiss, or the lass has been the subject of village gossip (that she's out among the fields where she can be a bit free with her affections) -- and in the song she's asking why, just because she's given away a kiss or two, or the boys have smiled at her in the fields, should she cry or be ashamed?