We present a multifrequency electron spin resonance study in the range of 4-420 GHz of the quasi-one-dimensional, nondimerized, quarter-filled Mott insulators, delta-(EDT-TTF-CONMe2)(2)X (X=AsF6, Br). In the high-temperature orthorhombic phase above T similar to 190 K, the magnitude and the temperature dependence of the high-temperature spin susceptibility are described by a S=1/2 Heisenberg antiferromagnetic chain with J(AsF6)=298 K and J(Br)=474 K coupling constants for X=AsF6 and Br, respectively. We estimate from the temperature dependence of the linewidth (Delta H) an exchange anisotropy, J'/J of similar to 2 X 10(-3). The frequency dependence of Delta H and the g shift have an unusual quadratic dependence in all crystallographic orientations that we attribute to an antisymmetric exchange (Dzyaloshinskii-Moriya) interaction.