We present the design of negative-refractive-index acoustic metamaterials operating at near-megahertz frequencies, intended for the eventual aim of enabling enhanced acoustic transmission through highimpedance-contrast biological layers. Leveraging the concept of complementary acoustic metamaterials, the negative effective properties of the metamaterials are designed to match the magnitude of an ultrasound-blocking, high-impedance-contrast layer's properties. The negative properties are obtained using a linear array of unit cells containing Helmholtz resonators and membranes. Using finite-element and analytical models, we calculate the band structure and the effective modulus and density of the proposed negative metamaterials. Using the full three-dimensional model of the metamaterials, we then simulate the enhancement of ultrasound transmission, through layers with high-impedance contrast to water. For instance, we see an improvement from 80% transmission through a high-impedance layer alone to 98% transmission through the metamaterial-plus-high-impedance layer combination. Scaling arguments are provided to estimate the system dimensions needed to provide higher operational frequencies appropriate for imaging and high-intensity ultrasound applications. Finally, as a proof of feasibility, a preliminary experimental realization of the unit-cell structure, created using the nanofabrication approach, is presented and tested in the near-megahertz regime. These results provide a step toward metamaterial-enhanced devices for noninvasive biomedical ultrasonic imaging and therapy.