HD 3167 is a bright (V = 8.9), nearby K0 star observed by the NASA K2 mission (EPIC 220383386), hosting two small, short-period transiting planets. Here we present the results of a multi-site, multi-instrument radial velocity campaign to characterize the HD 3167 system. The masses of the transiting planets are 5.02+/-0.38 MEarth for HD 3167 b, a hot super-Earth with a likely rocky composition (rho_b = 5.60+2.15-1.43 g/cm^3), and 9.80+1.30-1.24 MEarth for HD 3167 c, a warm sub-Neptune with a likely substantial volatile complement (rho_c = 1.97+0.94-0.59 g/cm^3). We explore the possibility of atmospheric composition analysis and determine that planet c is amenable to transmission spectroscopy measurements, and planet b is a potential thermal emission target. We detect a third, non-transiting planet, HD 3167 d, with a period of 8.509+/-0.045 d (between planets b and c) and a minimum mass of 6.90+/-0.71 MEarth. We are able to constrain the mutual inclination of planet d with planets b and c: we rule out mutual inclinations below 1.3 degrees as we do not observe transits of planet d. From 1.3-40 degrees, there are viewing geometries invoking special nodal configurations which result in planet d not transiting some fraction of the time. From 40-60 degrees, Kozai-Lidov oscillations increase the systems instability, but it can remain stable for up to 100Myr. Above 60 degrees, the system is unstable. HD 3167 promises to be a fruitful system for further study and a preview of the many exciting systems expected from the upcoming NASA TESS mission.
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