What 3 techniques does Tan's writing rely on heavily?
I can tell you about two of her techniques: structure and narrative.
Tan uses the "talk story" form of narrative. This is a Chinese "folk art form by which characters pass on values and teach important lessons through narrative" (from Salem Press, linked below). Although the stories that make up The Joy Luck Club are all told in the first-person point of view, there are multiple speakers; half of the stories are told by the mothers and half by their daughters.
The technique Tan uses to hold the stories together is called cradling or nesting, "a formal literary device that can be thought of as telling a story within a story." The eNotes discussion of the themes of the book continues:
Tan embeds the daughters' stories within the mothers' narratives. The Joy Luck Club is divided into four main segments that contain sixteen stories. The first and last sections tell eight stories—two for each mother—while the middle two sections each tell a story for each of the four daughters.
I'm not sure you would call it a technique, but Tan sets the novel in two different eras: the past of the mothers and the present of the daughters.