Though it does involve a pile of reading.
Say, Andrew Lang's books -- The Blue Fairy Book, The Red Fairy Book, etc. -- twelve in all. Grimms' collection. Joseph Jacob's -- between English, Celtic, Indian, and European, six in all. Asbjørnsen and Moe.
Lots of them in libraries. Or, since they are public domain, you can check them out here. Note that I do not particularly recommend Perrault, or Anderson, or Madame d'Aulnoy, or Giambattista Basile or Giovanni Francesco Straparola for this purpose. They are the literary variants. The others, on the other hand. . . . Thomas Crane for Italian (or Italo Calvino, but you'll have to buy it or getting from the library, it's not public domain) or A. H. Wratislaw for Slavonic tales. . . .
And other, not public domain ones. Some cultures they weren't published until too late for public domain, and others weren't translated in time. But after you've read two or three dozen such collections, you will have a feel for commonalities. You can construct your own variant, stealing the basic plot, and flourishes from this tale and that one, plus a few of your own devising.
If you already know you want a tale, you can check out more of Sur La Lune: all the fairy tales in the left bar have a selection of variants among their other annotations, so you can pick out "The Six Swans" and read a dozen or so variants and get a feel for how it changes cross cultures.