Context. This paper is a follow-up on the vast effort to collect radial velocity data for stars belonging to the Hipparcos survey. Aims. We aim at extending the orbital data available for binaries with M giant primaries. The data presented in this paper 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). Methods. Keplerian solutions are fitted to radial-velocity data. However, for several stars, no satisfactory solution could be found, even though the radial-velocity standard deviation is greater than the instrumental error, because M giants suffer from intrinsic radialvelocity 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. Results. We present an extensive collection of spectroscopic orbits for M giants with 12 new orbits, plus 17 from the literature. On top of these, 1 preliminary orbit yielded an approximate value for the eccentricity and the orbital period. Moreover, to illustrate how 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 115 521, 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. Conclusions. The data presented in this paper considerably increase the orbital data set for M giants, and will allow us to conduct a detailed analysis of the eccentricity -period diagram in a companion paper (Paper III).