Camera & Equipment
At one time, I would have said any camera that allows you to focus down to 1:1 with or without a macro lens, extension tubs, or supplementary lens will do, but now that point and shoot digital cameras and cell phone cameras are so popular, I am not so sure.
Point and Shoots and Cell Phone Cameras
These types of cameras tend to be short on features that allow good control of exposure and focus. Trying to see if a macro image on an LCD screen is in focus seems to be near impossible, and trying to control depth of field is no better. It may be that these are entry-level cameras and are generally used by people new to photography, thus they don't have the experience to use them well.
Digital and Film SLR
Even when these cameras do not have interchangeable lenses, they generally give you the control needed to select good focus and control over the depth of field. The ability to change lenses and therefore use a true macro lens is a real bonus in keeping your nose out of the mud! I personally use a Cannon 7D, chosen at the time because it was one of the few cameras with mirror lock and that could be controlled remotely. The technical side of digital cameras is moving at such a fast rate that I would not be surprised if there were even a more suitable camera available now..
Getting close
I started out using a standard 50 mm lens with extension tubes and close-up lenses. This worked fine, although it was a bit of a hassle having to change extension tubes all the time. When I updated my camera to Canon EOS3, I switched to a 100 mm macro lens, although a 50 or 60 mm lens would also do. The extra working distance is great for keeping your nose away from the ground.
Me taking a photoSo often I see people trying to hand hold a camera when photographing fungi. This is never going to work due to insufficient light. Regardless of whether you are using a SLR or a point and shoot camera to hand hold, you need a fast shutter speed. The only way you are going to get this is to use a large aperture and therefore no depth of field or a high ISO, giving you noise and grain.

A medium to heavy tripod is needed. One that will let the camera sit right down on the ground is even better. I use an Unlock 1700 tripod with a quick release bull head. Many find this type of tripod too heavy and go for something lighter. I really have problems with camera shake with my heavy tripod, but lugging this around the bush is not fun. I have tried to use a light-weight plastic tripod but found it would sit on top of leaf litter and bounce around too much. 
Shutter release
A remote shutter release, wired or wireless, also helps to remove any possibility of the camera shaking when you operate the shutter. Alternatively, the self-timer that most cameras have can be used. If your camera has a mirror lock, then use it too!
As mentioned elsewhere, try to choose a point of view where the light is coming in from the side. If it's from behind you, then you'll end up working in your own or the camera's shadow. making use of several reflectors to balance the light or to get light under the fungus and on its stalk. These can be as simple as white cards or commercial products.
Using flash
A while back, I invested in a flash unit (MT-24EX) for my camera. This flash has two heads and you can control the power level for each, thus giving you a shadow effect. The results are interesting but not what I would have prefered and I don't find flash works that well with fungi.
I have over time lost a few lenses from water damage and mould growing inside the lenses. Due to me and the camera being caught out in the rain or using the camera in damp locations. These days, I store my camera in a humidity controlled cupboard. This helps to dry out the camera and camera pack and reduces the chance of mould growing in the lens.