We report the design and performance of 300-mK composite bolometers that use micromesh absorbers and support structures patterned from thin films of low-stress silicon nitride. The small geometrical filling factor of the micromesh absorber provides 20x reduction in heat capacity and cosmic ray cross section relative to a solid absorber with no loss in IR-absorption efficiency. The support structure is mechanically robust and has a thermal conductance, G < 2 x 10(-11) W/K, which is four times smaller than previously achieved at 300 mK. The temperature rise of the bolometer is measured with a neutron transmutation doped germanium thermistor attached to the absorbing mesh. The dispersion in electrical and thermal parameters of a sample of 12 bolometers optimized for the Sunyaev-Zel'dovich Infrared Experiment is +/-7% in R (T), +/-5% in optical efficiency, and +/-4% in G.