A method is for making an electronic device that includes a multilayer circuit board having a non-planar three-dimensional shape defining a membrane switch recess therein. The multilayer circuit board may include at least one liquid crystal polymer (LCP) layer, and at least one electrically conductive pattern layer thereon defining at least one membrane switch electrode adjacent the membrane switch recess to define a membrane switch. The electronic may further include a compressible dielectric material filling the membrane switch recess. The electronic device may also include at least one spring member within the membrane switch recess.