In 2-quart saucepan over medium to low heat, place green chiles and cumin. Slowly add cream cheese and American cheese, gently stirring and cooking until melted. Pour queso into warming dish or small slow cooker set on low heat setting to keep warm.
Heat oven to 375°F. Unroll dough onto lightly floured work surface. Separate dough into breadsticks. Cut each breadstick in half crosswise and then in half lengthwise to make 4 strips.
Sprinkle cheese and oregano on each dough strip; gently press cheese and oregano into dough. Twist each strip; place on ungreased cookie sheet.
Bake 11 to 13 minutes or until brown and crisp. Serve breadsticks with warm dip.