In an effort to measure the masses of planets discovered by the NASA {\it K2} mission, we have conducted precise Doppler observations of five stars with transiting planets. We present the results of a joint analysis of these new data and previously published Doppler data. The first star, an M dwarf known as K2-3 or EPIC~201367065, has three transiting planets ("b", with radius $2.1~R_{⊕}$; "c", $1.7~R_{⊕}$; and "d", $1.5~R_{⊕}$). Our analysis leads to the mass constraints: $M_{b}=8.1^{+2.0}_{-1.9}~M_{⊕}$ and $M_{c}$ < $ 4.2~M_{⊕}$~(95\%~conf.). The mass of planet d is poorly constrained because its orbital period is close to the stellar rotation period, making it difficult to disentangle the planetary signal from spurious Doppler shifts due to stellar activity. The second star, a G dwarf known as K2-19 or EPIC~201505350, has two planets ("b", $7.7~R_{⊕}$; and "c", $4.9~R_{⊕}$) in a 3:2 mean-motion resonance, as well as a shorter-period planet ("d", $1.1~R_{⊕}$). We find $M_{b}$= $28.5^{+5.4}_{-5.0} ~M_{⊕}$, $M_{c}$= $25.6^{+7.1}_{-7.1} ~M_{⊕}$ and $M_{d}$ < $14.0~M_{⊕} $~(95\%~conf.). The third star, a G dwarf known as K2-24 or EPIC~203771098, hosts two transiting planets ("b", $5.7~R_{⊕}$; and "c", $7.8~R_{⊕}$) with orbital periods in a nearly 2:1 ratio. We find $M_{b}$= $19.8^{+4.5}_{-4.4} ~M_{⊕}$ and $M_{c}$ = $26.0^{+5.8}_{-6.1}~M_{⊕}$.....