Angular distributions of the 156, 158, 160, 162, 164Dy(t, p) reactions have been measured using 17 MeV tritons from the McMaster University Tandem Van de Graaff accelerator. The reaction products were analyzed with a magnetic spectrograph and detected with photographic emulsions, resulting in overall peak widths of 15–20 keV (FWHM). Levels populated with L=0 transitions were identified from the unambiguous angular distributions, and at least one previously-unknown Iπ = 0+ state was found in each nuclide studied. New Iπ=0+ levels were found at energies of 1269, 1549, 1743, and 2000 keV in 158Dy, 1457 and 1709 keV in 160Dy, 2126 keV in 162Dy, 1774 keV in 164Dy, and 1149 keV in 166Dy. Also, the previously proposed 0+ assignment for the 1655.4 keV level in 164Dy has been confirmed. For the neutron-rich nuclide 166Dy there was previously no information on excited states listed in the Nuclear Data Sheets, and many levels have been located in the present study, including the gamma band and an excited Kπ=0+ band. The 1457 keV 0+ state in 160Dy is a possible candidate for the bandhead of the previously reported S-band. The fraction of the L=0 strength which feeds excited states is unsually high in several cases, and particularly for 164Dy, in which the total L=0 strength to excited levels is ∼35% of that for the ground state. This can be explained qualitatively in terms of a “sub-shell closure” corresponding to the gap in the Nilsson diagram at N=98, and supports the earlier explanation of large L=0 strengths to excited states in the 161Dy(t, p) 163Dy reaction as being due to this gap. Most of the Lπ=0+ excited states observed in previous (p, t) studies were not populated in the present work, thus ruling out a pairing vibrational interpretation for these levels.