We present results from a campaign of multiple epoch echelle spectroscopy of relatively faint ( V = 9.5 - 13.5 mag ) red giants observed as potential astrometric grid stars for the Space Interferometry Mission ( SIM PlanetQuest ) . Data are analyzed for 775 stars selected from the Grid Giant Star Survey spanning a wide range of effective temperatures ( T _ { eff } ) , gravities and metallicities . The spectra are used to determine these stellar parameters and to monitor radial velocity ( RV ) variability at the 100 m s ^ { -1 } level . The degree of RV variation measured for 489 stars observed two or more times is explored as a function of the inferred stellar parameters . The percentage of radial velocity unstable stars is found to be very high – about 2/3 of our sample . It is found that the fraction of RV-stable red giants ( at the 100 m s ^ { -1 } level ) is higher among stars with T _ { eff } \sim 4500 K , corresponding to the calibration-independent range of infrared colors 0.59 < ( J - K _ { s } ) _ { 0 } < 0.73 . A higher percentage of RV-stable stars is found if the additional constraints of surface gravity and metallicity ranges 2.3 < \log~ { } g < 3.2 and -0.5 < [ Fe/H ] < -0.1 , respectively , are applied . Selection of stars based on only photometric values of effective temperature ( 4300 K < T _ { eff } < 4700 K ) is a simple and effective way to increase the fraction of RV-stable stars . The optimal selection of RV-stable stars , especially in the case when the Washington photometry is unavailable , can rely effectively on 2MASS colors constraint 0.59 < ( J - K _ { s } ) _ { 0 } < 0.73 . These results have important ramifications for the use of giant stars as astrometric references for the SIM PlanetQuest .