If I think of a checkerboard, is there a similar structure to see in my visual cortex? When I´m imagining a checkerboard in my mind, can one find, with all kinds of fancy equipment, a corresponding figure in the area of my visual cortex? For example, the neurons firing in such a way as they do when actually seeing a board, in which case the firing neurons obvious have the shape of a chessboard. 

When I've seen a lot of chessboards I can imagine one by the same pattern of firing neurons (which have for this particular case all different, or more or less the same connection strengths) as in the actual case of seeing one.
Probably. What you mentioned in your question is called retinotopy. There is a mapping between locations on your retina and areas on your cortex. As you go further up the visual processing streams, the mapping gets more complex and the patterns would be less obvious.

Here's an image of from a 1988 paper in the Journal of Neuroscience (Tootell, et al.). It shows a visual pattern on the left, and the corresponding neurons that were activated by that pattern in a monkey's primary visual cortex. The pattern was created using a technique called Deoxyglucose Labeling, where active neurons absorb a radioactive substance which can later be detected during dissection.



Also, Here's something similar in a mouse brain using a different technique.

Your question is about imagining a visual image, though. That is more complicated and research is currently underway to answer that question. Evidence points towards similar cortical patterns between imagined and perceived stimuli. Here is a study, not as detailed as the mouse or monkey ones, that found some evidence for retinotopy during imagining visual stimuli. Here is another where researchers could decode an imagined visual stimulus through fMRI. And here is yet another using fMRI to detect the direction of imagined motion.

There is a huge difference between the precise results from the animal perception studies and the human imagining studies. One big difference is that it is very difficult or impossible to do invasive studies with humans where you can get such high precision. Another difficulty with this research is knowing exactly when and what someone is imagining. It's easy to correlate brain activity with an external stimulus, because you know exactly what is occurring. All you have for the imagining case is self-report, which can be flawed.

Anyway, the research is progressing and there will be a good answer to this question soon enough.