HD 3167 is a bright (V = 8.9), nearby KO 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-circle plus for HD 3167 b, a hot super-Earth with a likely rocky composition (rho(b) = 5.6(-1.43)(+2.15) g cm(-3)), and 9.80(-1.24)(+1.30) M-circle plus 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-circle plus. We are able to constrain the mutual inclination of planet d with planets b and c: we rule out mutual inclinations below 1.degrees 3 because we do not observe transits of planet d. From 1.degrees 3 to 40 degrees, there are viewing geometries invoking special nodal configurations, which result in planet d not transiting some fraction of the time.