Context. The Ca II H line is one of the strongest lines in the solar spectrum, and it provides continuous information on the solar atmosphere from the photosphere to the lower chromosphere. Aims: We describe an inversion approach that reproduces observed Ca II H spectra by assuming local thermal equilibrium (LTE). Methods: We developed an inversion strategy based on the SIR code that reproduces Ca II H spectra in the LTE approximation. The approach uses a two-step procedure with an archive of pre-calculated spectra to fit the line core and a subsequent iterative modification to improve the fit mainly in the line wing. Simultaneous spectra in the 630 nm range can optionally be used to fix the continuum temperature. The method retrieves one-dimensional (1D) temperature stratifications while neglecting lateral radiative transport. Line-of-sight velocities are included post facto with an empirical approach. Results: An archive of about 300 000 pre-calculated spectra is more than sufficient to reproduce the line core of observed Ca II H spectra both in the quiet Sun and in active regions. The subsequent iterative adjustment of the thermodynamical stratification matches observed and best-fit spectra to a level of about 0.5% of Ic in the line wing and about 1% of Ic in the line core. Conclusions: The successful application of the LTE inversion strategy suggests that inversion schemes based on pre-calculated spectra allow a reliable and relatively fast retrieval of solar properties from observed chromospheric spectra. The approach can be easily extended to a 1D non-LTE (NLTE) case by a simple exchange of the pre-calculated archive spectra. Using synthetic NLTE spectra from numerical three-dimensional (3D) simulations instead will finally allow one to extend the approach from the static 1D-case to dynamical atmosphere models, including the complete 3D radiative transport. The animation is available in electronic form at http://www.aanda.org