Triangle mesh maps are a versatile 3D environment representation for robots to navigate in challenging indoor and outdoor environments exhibiting tunnels, hills and varying slopes. To make use of these mesh maps, methods are needed to accurately localize robots in such maps to perform essential tasks like path planning and navigation. We present Mesh ICP Localization (MICP-L), a novel and computationally efficient method for registering one or more range sensors to a triangle mesh map to continuously localize a robot in 6D, even in GPS-denied environments. We accelerate the computation of ray casting correspondences (RCC) between range sensors and mesh maps by supporting different parallel computing devices like multicore CPUs, GPUs and the latest NVIDIA RTX hardware. By additionally transforming the covariance computation into a reduction operation, we can optimize the initial guessed poses in parallel on CPUs or GPUs, making our implementation applicable in real-time on many architectures. We demonstrate the robustness of our localization approach with datasets from agricultural, aerial, and automotive domains.