Lagrangian properties of oceanic turbulent boundary layers were measured using neutrally buoyant floats. Vertical acceleration was computed from pressure (depth) measured on the floats. An average vertical vorticity was computed from the spin rate of the float. Forms for the Lagrangian frequency spectra of acceleration, ϕa(ω), and the Lagrangian frequency spectrum of average vorticity are found using dimension analysis. The flow is characterized by a kinetic energy dissipation rate, ε, and a large-eddy frequency, ω0. The float is characterized by its size. The proposed non-dimensionalization accurately collapses the observed spectra into a common form. The spectra differ from those expected for perfect Lagrangian measurements over a substantial part of the measured frequency range owing to the finite size of the float. Exact theoretical forms for the Lagrangian frequency spectra are derived from the corresponding Eulerian wavenumber spectra and a wavenumber–frequency distribution function used in previous numerical simulations of turbulence. The effect of finite float size is modelled as a spatial average. The observed non-dimensional acceleration and vorticity spectra agree with these theoretical predictions, except for the high-frequency part of the vorticity spectrum, where the details of the float behaviour are important, but inaccurately modelled. A correction to the exact Lagrangian acceleration spectra due to measurement by a finite-sized float is thus obtained. With this correction, a frequency range extending from approximately one decade below ω0 to approximately one decade into the inertial subrange can be resolved by the data. Overall, the data are consistent with the proposed transformation from the Eulerian wavenumber spectrum to the Lagrangian frequency spectrum. Two parameters, ε and ω0, are sufficient to describe Lagrangian spectra from several different oceanic turbulent flows. The Lagrangian Kolmogorov constant for acceleration, βa≡ϕa/ε, has a value between 1 and 2 in a convectively driven boundary layer. The analysis suggests a Lagrangian frequency spectrum for vorticity that is white at all frequencies in the inertial subrange and below, and a Lagrangian frequency spectrum for energy that is white below the large-eddy scale and has a slope of −2 in the inertial subrange.