 The Markovian master equation provides an approximate tool for describing open quantum systems, but it often fails to respect thermodynamics without a secular approximation. We introduce a new equation that is thermodynamically consistent and accurate when memory effects can be neglected by rescaling the Hamiltonian for bookkeeping and exploiting the limited resolution of heat in a Markovian description. This enables a thermodynamically consistent description of various systems where the secular approximation fails. This article was authored by Patrick Peapots, Alex Orosh Sandkali, and Andreas Wacker.