We present APEX 218GHz observations of molecular emission in a complete sample of embedded protostars in the Ophiuchus star-forming region. To study the physical properties of the cores, we calculate H_2_CO and c-C_3_H_2_ rotational temperatures, both of which are good tracers of the kinetic temperature of the molecular gas. We find that the H_2_CO temperatures range between 16K and 124K, with the highest H_2_CO temperatures toward the hot corino source IRAS 16293-2422 (69-124K) and the sources in the {rho} Oph A cloud (23-49K) located close to the luminous Herbig Be star S1, which externally irradiates the {rho} Oph A cores. On the other hand, the c-C_3_H_2_ rotational temperature is consistently low (7-17K) in all sources. Our results indicate that the c-C_3_H_2_ emission is primarily tracing more shielded parts of the envelope whereas the H_2_CO emission (at the angular scale of the APEX beam; 3600 au in Ophiuchus) mainly traces the outer irradiated envelopes, apart from in IRAS 16293-2422, where the hot corino emission dominates. In some sources, a secondary velocity component is also seen, possibly tracing the molecular outflow.