X-ray telescopes are observing the x-rays emitted from matter as it heats during its activity near a black hole. X-ray observatories have been taking images of supermassive black holes including images of Sagittarius A*, and the observations have been fruitful, although they have not yet produced a definitive image of the black hole silhouette (see below). What the observations of Sagittarius A* have told us is that the black hole we believe is at the center of our galaxy has shown erratic behavior in the form of flaring and dimming. This action is taking place near the event horizon, and accounts for the relatively low amount of radiation emission coming from Sagittarius A*. This black hole is estimated to have 4 millions times the mass of our sun, and yet the radiation emitted is billions of times weaker than the radiation coming from other black holes.
The flaring was observed by an x-ray telescope utilizing a process called light echo where the telescope observes changes caused by x-ray pulses from Sagittarius A* in a cloud nearby, called Sagittarius B2. The process can better be described as follows: "When the x-rays reach the cloud [Sagittarius B2], they collide with iron atoms, kicking out electrons that are close to the atomic nucleus. When electrons from farther out fill in these gaps, the iron atoms emit x-rays. But after the X-ray pulse passes through, the cloud fades to its normal brightness" (http://chandra.harvard.edu/press/08_releases/press_041608.html). However nearby, the x-ray pulses observed took about 300 years to reach Sagittarius B2, so the information recorded from there is 3 centuries old, and the information we receive from Sagittarius B2 takes about 26,000 years to reach us on Earth (that being the approximate distance to the center of our galaxy, with the x-rays traveling at the speed of light).
The Chandra X-ray Observatory is one tool being used prominently in the effort to image black holes. Chandra is a NASA - Smithsonian Astrophysical Observatory satellite in high orbit around Earth, actually orbiting at over one-third the distance from Earth to the Moon. There are also several x-ray telescopes operated by the Japanese Aerospace Exploration Agency (the newest being Suzaku, launched in 2005, older ones being Hinotori, Tenma, Ginga and Asuka) searching for x-ray emissions from astronomical bodies.