The relationship between anisotropic electronic circular dichroism (CD) and the molecular structure for planar cis-butadiene and for 1,3-cyclohexadiene and some of its allylic methyl derivatives is studied through calculation and graphical display of the rotatory strength tensor for the lowest pi --> pi* excitation in these systems. Also included is cis-butadiene in chiral structures mimicking the diene units in two of the cyclohexadiene systems. The calculations are done ab initio in the random phase approximation using an aug-cc-pVTZ atomic basis set chosen from a systematic basis set study for chiral cis-butadiene. For planar cis-butadiene this study provides the first calculation of anisotropic CD of an achiral molecule and predicts a CD intensity distribution exhibiting two numerically equal, but oppositely signed, lobes along mutually orthogonal directions perpendicular to the C-2 axis for the system. The CD intensities for the chiral molecules cyclohexadiene and its allylic derivatives exhibit two large and oppositely signed major lobes, echoing the CD of cis-butadiene, in addition to a nonvanishing CD intensity along the C-2 axis of the diene unit. The chirality of the ring conformation and of the arrangement of the substituents is reflected in a difference between the magnitudes of the two major lobes in the CD response and in the variation in sign and magnitude of the CD intensity along the C-2 axis. More specifically, the effects of the allylic methyl groups follow a quadrant rule and are almost additive, the effects being significantly larger for axial than for equatorial substituents. The helical twisting of the diene chromophore is of minor importance. These trends are found also for the isotropic CD, confirming earlier results. The analysis of the anisotropic CD suggests that the CD intensity along the uniaxial C-2 direction of the diene may serve as a sensitive chiral indicator for these systems.