Different from the Einstein's general relativity which describes gravity through the curvature of a spacetime, teleparallel gravity theory depicts gravity through the Weitzenbock torsion of the spacetime. This leads to different expressions of the dynamical equations of fundamental fields, including Dirac fields, in both theories. Here we derive two types of solutions of the Dirac's equations for the case of isotropic Bianchi type I spacetime. The first is an oscillatory solution, i.e. the time dependence of the solution is chosen to be sinusoidal. Here we derive the general form of space coordinate dependent part of the solution. The second type of solution is a solution where the space coordinate part of solution is chosen to be sinusoidal. Here we derive the time dependence of the solution for the case of exponential scale factor.