We introduce entropy convergence rates as isomorphism invariants for measure-preserving systems and prove several general facts concerning these rates for aperiodic systems, completely ergodic systems and rank-one systems. We will for example show that for any completely ergodic system $(X,T)$ and any non-trivial partition $\alpha$ of $X$ into two sets we have $\limsup_{n\rightarrow\infty}H(\alpha_0^{n-1})/g(\log_2n)=\infty$, whenever $g$ is a positive increasing function on $(0,\infty)$ such that $g(x)/x^2$ is integrable.