Kramers-Kronig (KK) receivers have been shown to be able to cancel the signal-signal beat interference that occurs upon direct-photodetection in optical single-sideband systems, where the transmitted optical field is proportional to the modulation signal (Field-Modulated Direct-Detection systems). They can support complex modulation formats, and being singlesideband, electronic dispersion compensation, without the need for a coherent receiver. All current research has used digital-signal processing (DSP) to implement the KK algorithm.Radio-frequency KK receivers using analog processing were first introduced in the 1960's, and offer an alternative to DSP, particularly in their approximate form. Analog processing is inherently real-time and offers the lowest latency (processing delay). In this paper, we show that with minor changes, analog processing can also be used with signals from a photodetector. We have built an experimental receiver, using standard parts that have functionality that approximates to the desired algorithm. In particular, we show that a differential pair of bipolar transistors can approximate a square-root function, that the Hilbert transform can be approximated by an 8-tap transmission-line delay, and that a commercial analog multiplier chip can be used for a squaring and subtraction function.The system is demonstrated with 16-QAM at 500 Mbit/s with the photocurrent emulated in software and the KK algorithm performed in hardware. Finally, software band-limits the signal, down-converts it to form a constellation, and calculates symbol error rates. We show that the analog KK reduces the symbol error rate by at least a factor of ten, enabling low carrier-to-signal power ratios to be used with standard forward error correction. Much higher data rates could be supported using customdesigned microwave ASICs, which could be incorporated in packaged receivers. This would enable miniaturized receivers using standard DSP to work with dispersion-limited links. Interestingly, half the benefit could be gained by simply re-biasing the differential stage of a standard photoreceiver.