Reverse transcription polymerase chain reaction (RT-PCR) is the gold standard for the molecular diagnosis of many infectious diseases, including RNA viruses, but is generally limited to settings with access to trained personnel and laboratory resources. We have previously reported a fundamentally simpler thermal cycling platform called Adaptive PCR, which dynamically controls thermal cycling conditions during each cycle by optically monitoring the annealing and melting of mirror-image L-DNA surrogates of the PCR primers and targets. In this report, we integrate optically-controlled reverse transcription and single-channel monitoring of L-DNAs to develop a multiplexed Adaptive RT-PCR instrument and assay for the detection of Zika, dengue, and chikungunya virus RNA with high target specific and low limits of detection. The assay is demonstrated to detect as low as 5 copies/reaction of Zika or chikungunya RNA and 50 copies/reaction of dengue RNA. The multiplexed Adaptive RT-PCR instrument is robust and has many of the features required to implement diagnostic assays for RNA viruses in settings that lack traditional laboratory resources.