Periodic structures with extremely low porosities capable of forming large band gaps-frequency ranges with strong wave attenuation-are designed by patterning an elastic sheet with an array of alternating crack-like pores separated by small ligaments. The results indicate that the presence and size of the band gaps are controlled by the smallest geometric feature in the system (which can be easily controlled by tuning the aspect ratio of the pores), providing an important guideline for the design of systems with the desired response.