Efficient simulation of plasmas in various contexts often involves the use of meshes that conform to the intrinsic geometry of the system under consideration. We present here a description of a new magnetohydrodynamic code, GAMERA (Grid Agnostic MHD for Extended Research Applications), designed to combine geometric flexibility with high-order spatial reconstruction and constrained transport to maintain the divergence-free magnetic field. GAMERA carries on the legacy of its predecessor, the LFM (Lyon-Fedder-Mobarry), a research code whose use in space physics has spanned three decades. At the time of its initial development, the LFM code had a number of novel features: eighth-order centered spatial differencing, the Partial Donor Cell Method limiter for shock capturing, a non-orthogonal staggered mesh with constrained transport, and conservative averaging-reconstruction for axis singularities. The capability to handle multiple ion species was also added later. GAMERA preserves the core numerical philosophy of LFM while also incorporating numerous algorithmic and computational improvements. The upgrades in the numerical schemes include accurate grid metric calculations using high-order Gaussian quadrature techniques, high-order upwind reconstruction, non-clipping options for interface values, and improved treatment of axis singularities. The improvements in the code implementation include the use of data structures and memory access patterns conducive to aligned vector operations and the implementation of hybrid parallelism, using MPI and OMP. GAMERA is designed to be a portable and easy-to-use code that implements multidimensional MHD simulations in arbitrary non-orthogonal curvilinear geometries on modern supercomputer architectures.