electronic band theory - Density of States in NOT Free Electron Gas

I think that I understand how the density of states works for a free electron gas. It is effectively just a conversion factor between summing over values of k and integrating over values of E. If you want, you can look at it as the density of points along the k-axis of a discrete plot of the electronic dispersion.

From there, you can compute most thermodynamic quantities, like internal energy and heat capacity, along with their temperature dependencies.

However, once we move beyond the simple model, I don't really know what to do. (i.e. we can calculate the dispersion in the Kronig-Penney model. Is it useful to re-calculate the density of states for this? Would it even look different than the free electron gas density of states, given the potential has no k-dependence? Do we gain any physical insight by calculating the density of states when we have band theory--beyond that higher bands have more states in 3D, the same number of states in 2D, and fewer states in 1D--assuming the free electron gas density of states holds?)

Thanks!