I live in the United States and would say "I used to walk on rainy days" or "I used to take a walk on rainy days".

There may be regional differences as far as what is considered correct, but i would say: "I used to walk (take walks) on rainy days."

"On a rainy day" or "on rainy days" is standard English. This is the construction which you should use. What you heard could be a regional variation. You need to realise that native speakers from different parts of the world may speak different dialects of English, with slight variations in grammar.

An Irish speaker may speak a sort of hybrid Gaelic-Englis dialect. Or possibly Celtic-English hybrid grammar. They may do what we all do know single words in a new language but construct them into a sentence using the grammar of their native language. They may use that construction in the Irish language (Gaelic)

standard English is 'walk in the rain' 'walking in the rain' 'walk on a rainy day'. So it is on when specifying a day either any day or a specific day. In when specifying just being in the rain. 'I like going out on rainy days' 'I like being in the rain' in the UK you will often hear more ancient grammar used by older people, especially in remote rural areas. with a local dialect.