We consider semilinear parabolic equations of the form
$$
  u_t=u_{xx}+f(u),\quad x\in R,t>0,
$$
where  $f$ is a $C^1$ function. Assuming that $0$ and $\gamma>0$ are constant steady states, we investigate the large-time behavior of the front-like solutions, that is, solutions $u$ whose initial values  $u(x,0)$ are near $\gamma$ for $x\approx -\infty$ and near $0$ for $x\approx \infty$. If the steady states $0$ and $\gamma$ are both stable, our main theorem shows that at large times, the graph of  $u(\cdot,t)$ is arbitrarily close to a propagating terrace  (a system of stacked traveling fonts).  We prove this result without requiring  monotonicity of  $u(\cdot,0)$ or the nondegeneracy of zeros of $f$. The case when one or both of the steady states $0$, $\gamma$ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their $\omega$-limit sets with respect to the locally uniform convergence consist of steady states. In our proofs we employ phase plane analysis, intersection comparison (or, zero number) arguments,  and a geometric method involving the spatial trajectories $\{(u(x,t),u_x(x,t)):x\in R\}$, $t>0$, of the solutions in question.