The goal of magnetic resonance electrical impedance tomography (MREIT) is to produce a tomographic image of a conductivity distribution inside an electrically conducting object. Injecting current into the object, we measure a z-component B_z of an induced magnetic flux density using an MRI scanner. Based on the relation between the conductivity and measured B_z data, we may reconstruct cross-sectional images of the conductivity distribution. In a two-dimensional imaging slice, we can see that conductivity changes along equipotential lines are determined by the measured B_z data. Since the equipotential lines themselves depend nonlinearly on the unknown conductivity distribution, it is not possible to recover the conductivity distribution directly from measured B_z data. Conductivity image reconstruction algorithms such as the harmonic B_z algorithm utilize an iterative procedure to update conductivity values and in each iteration we need to solve an elliptic boundary value problem with a presumed conductivity distribution. This iteration is often troublesome in practice due to excessive amounts of noise in some local regions where weak MR signals are produced. This motivated us to develop a non-iterative reconstruction algorithm which does not depend on a global structure of the conductivity distribution. In this paper, we propose a new MREIT conductivity image reconstruction algorithm, called the non-iterative harmonic B_z algorithm, which provides a conductivity image directly from measured B_z data. From numerical simulations, we found that the new method produces absolute images of a conductivity distribution with a high accuracy by maximizing the use of reliable B_z data. It effectively reduces adverse effects of excessive noise in some local regions of weak MR signals by restricting the influences within them.