Characteristics of high-power, narrow-linewidth, continuous-wave (cw) green radiation obtained by simple single-pass second-harmonic-generation (SHG) of a cw ytterbium fiber laser at 1064 nm in the nonlinear crystals of PPKTP and MgO:sPPLT are studied and compared. Temperature tuning and SHG power scaling up to nearly 10 W for input fundamental power levels up to 30 W are performed. Various contributions to thermal effects in both crystals, limiting the SHG conversion efficiency, are studied. Optimal focusing conditions and thermal management schemes are investigated to maximize SHG performance in MgO:sPPLT. Stable green output power and high spatial beam quality with M(2)<1.33 and M(2)<1.34 is achieved in MgO:sPPLT and PPKTP, respectively.