The simplest way is to do it in cartesian coordinates: take x, y, and z independently from N(0,1). If you want to generate only one normal number per step, consider the jacobian in the angles.
Actually, this is wrong, as it would allow displacements (at 1 sigma) of 1 along the axis, but up to sqrt(3) along diagonals. What you actually want is a multivariate normal distribution with covariance proportional to the identity (uncorrelation between axis and isotropy).