We study the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, where $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d)})^T$ and $Z_t^{(1)}, \ldots, Z_t^{(d)}$ are independent one-dimensional L{é}vy processes with characteristic exponents $ψ_1, \ldots, ψ_d$. We assume that each $ψ_i$ satisfies a weak lower scaling condition WLSC($α,0,\underline{C}$), a weak upper scaling condition WUSC($β,1,\overline{C}$) (where $0< α\le β< 2$) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all $ψ_1, \ldots, ψ_d$ are the same and $α, β$ are arbitrary, or (ii) not all $ψ_1, \ldots, ψ_d$ are the same and $α> (2/3)β$. We also assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed $γ\in (0,α) \cap (0,1]$ the semigroup $P_t$ of the process $X$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. We also show the existence of a transition density of the process $X$.