  This paper is a follow-up of the vast effort to collect radial velocity data
for stars belonging to the Hipparcos survey. We aim at extending the orbital
data available for binaries with M giant primaries. The data will be used in
the companion papers of this series to (i) derive the binary frequency among M
giants and compare it to that of K giants (Paper II), and (ii) analyse the
eccentricity-period diagram and the mass-function distribution (Paper III).
Keplerian solutions are fitted to radial-velocity data. However, for several
stars, no satisfactory solution could be found, despite the fact that the
radial-velocity standard deviation is larger than the instrumental error,
because M giants suffer from intrinsic radial-velocity variations due to
pulsations. We show that these intrinsic radial-velocity variations can be
linked with both the average spectral-line width and the photometric
variability. We present an extensive collection of spectroscopic orbits for M
giants, with 12 new orbits, plus 17 from the literature. Moreover, to
illustrate the fact that the large radial-velocity jitter present in Mira and
semi-regular variables may easily be confused with orbital variations, we also
present examples of pseudo-orbital variations (in S UMa, X Cnc and possibly in
HD 115521, a former IAU radial-velocity standard). Because of this difficulty,
M giants involving Mira variables were excluded from our monitored sample. We
finally show that the majority of M giants detected as X-ray sources are
actually binaries.
