We have performed Joule power loss calculations for a flat dechirper. We have considered the configurations of the beam on-axis between the two plates---for chirp control---and for the beam especially close to one plate---for use as a fast kicker. Our calculations use a surface impedance approach, one that is valid when corrugation parameters are small compared to aperture (the perturbative parameter regime). In our model we ignore effects of field reflections at the sides of the dechirper plates, and thus expect the results to underestimate the Joule losses. The analytical results were also tested by numerical, time-domain simulations. We find that most of the wake power lost by the beam is radiated out to the sides of the plates. For the case of the beam passing by a single plate, we derive an analytical expression for the broad-band impedance, and---in Appendix B---numerically confirm recently developed, analytical formulas for the short-range wakes. While our theory can be applied to the LCLS-II dechirper with large gaps, for the nominal apertures we are not in the perturbative regime and the reflection contribution to Joule losses is not negligible. With input from computer simulations, we estimate the Joule power loss (assuming bunch charge of $300$~pC, repetition rate of $100$~kHz) is 21~W/m for the case of two plates, and 24~W/m for the case of a single plate. Show Partial Abstract