Recent experiments on the response of individual, long DNA strands to externally-applied tension and fluid shear suggest that semi-dilute, aqueous suspensions of these molecules should behave as viscoelastic fluids with an elastic relaxation time of seconds and a viscosity comparable to that of water. I present experimental observations of the convective flow produced by heating a horizontal layer of such viscoelastic fluids from below, in a long, narrow, annular geometry. The convection patterns take the form of spatially-localized standing and travelling waves which exhibit small amplitudes and extremely long oscillation periods. The threshold Rayleigh numbers for the onset of oscillations are lower than the value measured for steady convection in a Newtonian fluid in the same apparatus and exhibit a decreasing trend with increasing elastic relaxation time. This behavior agrees with the predictions of theories of the linear instability of viscoelastic convection. However, the low frequencies are in gross quantitative disagreement with these analyses.