An accurate method is presented for flexural vibrations of rhombic plates having all combinations of clamped and free edge conditions. The prime focus here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, clamped-free corners having obtuse angles of the rhombic plates. Accurate non-dimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough obtuse angle of 165$^{circ}$, so that a significant influence of clamped-free corner stress singularities may be understood.