A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of switches of directions is fixed. The explicit unconditional distributions are also obtained and are expressed in terms of Bessel functions. The governing equations are derived and given as products of D'Alembert operators. The limiting form of the equations is provided in the Euclidean space $\mathbb{R}^d$ and takes the form of a heat equation with infinitesimal variance $1/d$.