ATLAS electron and photon triggers covering transverse energies from 5 GeV to several TeV are essential to record signals for a wide variety of physics: from Standard Model processes to searches for new phenomena. During LHC Run 3 (expected to start in 2021 or 2022 until 2024), the main triggers used for those physics studies will be a single-electron trigger with transverse energy threshold around 25 GeV, and a diphoton trigger with thresholds at 25 and 35 GeV. Relying on those simple, general-purpose triggers is a robust trigger strategy, tested already in Run 2 (2015--2018), at a cost of slightly higher trigger output rates, rather than to use a large number of analysis-specific triggers. In preparation for Run 3 data-taking, the ATLAS detector is undergoing an upgrade of the hardware-based Level-1 calorimeter trigger and the trigger software is being migrated to the multi-threaded framework AthenaMT. The impact from these modifications on the electron and photon triggers as well as their projected performance in Run 3 is presented.