  This paper presents finite-velocity random motions driven by fractional
Klein-Gordon equations of order $\alpha \in (0,1]$. A key tool in the analysis
is played by the McBride's theory which converts fractional hyper-Bessel
operators into Erdelyi-Kober integral operators. Special attention is payed to
the fractional telegraph process whose space-dependent distribution solves a
non-homogeneous fractional Klein-Gordon equation. The distribution of the
fractional telegraph process for $\alpha = 1$ coincides with that of the
classical telegraph process and its driving equation converts into the
homogeneous Klein-Gordon equation. Fractional planar random motions at finite
velocity are also investigated, the corresponding distributions obtained as
well as the explicit form of the governing equations. Fractionality is
reflected into the underlying random motion because in each time interval a
binomial number of deviations $B(n,\alpha)$ (with uniformly-distributed
orientation) are considered. The parameter $n$ of $B(n,\alpha)$ is itself a
random variable with fractional Poisson distribution, so that fractionality
acts as a subsampling of the changes of directions. Finally the behaviour of
each coordinate of the planar motion is examined and the corresponding
densities obtained. Extensions to $N$-dimensional fractional random flights are
envisaged as well as the fractional counterpart of the Euler-Poisson-Darboux
equation to which our theory applies.
