We consider in this paper random flights in $\mathbb{R}^d$ performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on spheres $S_1^d$. For the position $(X_1(t),...,X_d(t))$ we obtain the conditional characteristic function $$E\left\{e^{i\sum_{k=1}^d \alpha_k X_k(t)}\mid N(t)=n\right\}$$ and related density $p_n(x_1,...,x_d;t)$ in terms of $(n+1)-$fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension $d=2$ and $d=4$. In these two cases also the unconditional distribution is determined in explicit form. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for $d=2$. The related motion with random velocity in $\mathbb{R}^3$ is analyzed and its distribution derived.