Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the results obtained imply the sharp Jackson-type inequality \(\| f-X_{n,r,\mu}(f)\| _{p}\leq \frac{K_r}{2n^r} \omega_1\left(f^{(r)},\frac{\pi}{n}\right)_p\).