A new method for dispersion mitigation is presented for low-cost and simple networks. The method does not require dispersion-compensating fibers, special optical filters, coherent detection, or external modulation. It can work in a direct modulation scheme and with a standard optical detector. The disadvantage of this method is the requirement to operate in the weak modulation regime of the signal. In this regime, the dispersive channel can be regarded as linear in the power domain, and not only in the field domain, so that the effects of dispersion can be reduced with a proper electronic filter. Since electronic filters are usually considerably cheaper than coherent optical solutions, this solution can be implemented in low-cost networks, where dispersion is a more severe problem than noise. We show that by adding the filter to a low-noise on-off keying (OOK) system it is possible to transmit data at bit rates of 50 Gb/s to distances at least sixfold larger than its OOK limit (6 km in this case), i.e., 40 km and beyond.