The one-dimensional structure of carbon nanotubes(1) leads to a variety of remarkable optical(2) and electrical(3) properties that could be used to develop novel devices(4). Recently, the electrical conductance of nanotubes has been shown to decrease under optically induced heating by an amount proportional to the temperature change(5). Here, we show that this decrease is also proportional to the initial nanotube conductance, and make use of this effect to develop a new electrical characterization tool for nanotubes. By scanning the focal spot of a laser across the surface of a device through which current is simultaneously measured, we can construct spatially resolved conductance images of both single and arrayed nanotube transistors. We can also directly image the gate control of these devices. Our results establish photothermal current microscopy as an important addition to the existing suite of characterization techniques for carbon nanotubes and other linear nanostructures.