Energy transfer effects on dicarbon (C2) d3Pi(g) <-- a3Pi(u) laser-induced fluorescence (LIF) spectra in fuel-rich acetylene atmospheric-pressure flames have been studied using a combination of two different two-dimensional techniques. Measurements using a picosecond laser system in conjunction with a streak camera allowed determination of typical fluorescence lifetimes of levels in the d state and of population changes introduced by rotational energy transfer (RET) and by state-dependent quenching. Excitation-emission spectroscopy yielded two-dimensional maps containing all excitation and all emission spectra in the spectral ranges between 19 340 and 20 150 cm(-1) (excitation) and from 546 to 573 nm (emission) and allowed unambiguous assignment of all transitions in this spectral region. Our measurements show a comparatively long quenching lifetime (around 2 ns) and dominant effects of energy transfer on shape and intensity of the acquired spectra (90% of the fluorescence stems from levels populated by ET). A pronounced dependence of the total RET on the quantum number of the initially excited state is observed. Vibrational energy transfer (VET) is significantly weaker (only 5% contribution for excitation in the v' = 1 level). Implications for quantitative concentration measurements are discussed, and exemplary spatially resolved profiles in a well-characterized low-pressure propene flame are presented.