Aims. We present new high-sensitivity 153 MHz Giant Meterwave Radio Telescope follow-up observations of the diffuse steep-spectrum cluster radio sources in the galaxy clusters Abell 521, Abell 697, and Abell 1682. Abell 521 hosts a relic, and together with Abell 697 it also hosts a giant very steep-spectrum radio halo. Abell 1682 is a more complex system with candidate steep spectrum diffuse emission. Methods. We imaged the diffuse radio emission in these clusters at 153 MHz, and provided flux density measurements of all sources at this frequency. Our new flux density measurements, coupled with the existing data at higher frequencies, allowed us to study the total spectrum of the halos and relic over at least one order of magnitude in frequency. Results. Our images confirm the very steep “diffuse component” in Abell 1682. We found that the spectrum of the relic in Abell 521 can be fitted by a single power-law with α = 1.45 ± 0.02 from 153 MHz to 5 GHz. Moreover, we confirm that the halos in Abell 521 and Abell 697 have a very steep spectrum , with α = 1.8 − 1.9 and α = 1.52 ± 0.05, respectively. Even including of the 153 MHz flux density information, it is impossible to distinguish between power-law and curved spectra, as derived from homogeneous turbulent re-acceleration. The latter are favoured on the basis of simple energetic arguments, and we expect that LOFAR will finally unveil the spectral shape of radio halos below 100 MHz, thus providing clues on their origin.