We present an architecture-portable and performant implementation of the atmospheric dynamical core (High-Order Methods Modeling Environment, HOMME) of the Energy Exascale Earth System Model (E3SM). The original Fortran implementation is highly performant and scalable on conventional architectures using the Message Passing Interface (MPI) and Open MultiProcessor (OpenMP) programming models. We rewrite the model in C++ and use the Kokkos library to express on-node parallelism in a largely architecture-independent implementation. Kokkos provides an abstraction of a compute node or device, layout-polymorphic multidimensional arrays, and parallel execution constructs. The new implementation achieves the same or better performance on conventional multicore computers and is portable to GPUs. We present performance data for the original and new implementations on multiple platforms, on up to 5400 compute nodes, and study several aspects of the single- and multi-node performance characteristics of the new implementation on conventional CPU (e.g., Intel Xeon), many core CPU (e.g., Intel Xeon Phi Knights Landing), and Nvidia V100 GPU.