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 M_{earth} for HD 3167 b, a hot super-Earth with a likely rocky composition ({rho}_b=5.60_-1.43_^+2.15^ g/cm^3^), and 9.80_-1.24_^+1.30^ M_{earth} for HD 3167 c, a warm sub-Neptune with a likely substantial volatile complement ({rho}_c=1.97_-0.59_^+0.94^ 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 M_{earth}_. We are able to constrain the mutual inclination of planet d with planets b and c: we rule out mutual inclinations below 1.3{deg} because we do not observe transits of planet d. From 1.3{deg} to 40{deg}, there are viewing geometries invoking special nodal configurations, which result in planet d not transiting some fraction of the time. From 40{deg} to 60{deg}, Kozai-Lidov oscillations increase the system's instability, but it can remain stable for up to 100 Myr. Above 60{deg}, 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.