Among the tens of thousands of known RR Lyrae stars there are only a handful that show indications of possible binarity. The question why this is the case is still unsolved, and has recently sparked several studies dedicated to the search for additional RR Lyraes in binary systems. Such systems are particularly valuable because they might allow to constrain the stellar mass. Most of the recent studies, however, are based on photometry by finding a light time effect in the timings of maximum light. This approach is a very promising and successful one, but it has a major drawback: by itself, it cannot serve as a definite proof of binarity, because other phenomena such as the Blazhko effect or intrinsic period changes could lead to similar results. Spectroscopic radial velocity measurements, on the other hand, can serve as definite proof of binarity. We have therefore started a project to study spectroscopically RR Lyrae stars that are suspected to be binaries. We have obtained radial velocity (RV) curves with the 2.1m telescope at McDonald observatory. From these we derive systemic RVs which we will compare to previous measurements in order to find changes induced by orbital motions. We also construct templates of the RV curves that can facilitate future studies. We also observed the most promising RR Lyrae binary candidate, TU UMa, as no recent spectroscopic measurements were available. We present a densely covered pulsational RV curve, which will be used to test the predictions of the orbit models that are based on the O-C variations.