Yes, the frames almost always have a negative effect. The simplest approach is to make them somewhat flush or thin and bevel the edges.
For smaller speakers, cutting them from one piece of 1/4" MDF can work pretty well when the inside and outside edges are beveled. There are many different approaches but I usually stay close to this one. aa
I made grills for the rear of my small open baffle speakers but since the fabric had to stretch around the back of the drivers, I used 1/2" MDF for added strength. I have no grills for the fronts on those.
Best of luck,
Bill