it's nothing to do with flippin' rainwater. They wouldn't have made 80 million open chimneys if it was.
Dampness on or around a chimney breast is likely to be caused by falling damp! Rainwater directly entering the chimney mouth and filtering down through the walls below. Capping your chimney could eliminate this problem.
http://www.maintainyourbuilding.org.uk/pages/getting_to_grips_with_damp.html
http://www.buildingconservation.com/articles/services/chimney.htm
There are two basic causes of damp in chimneys: ingress of rain, and condensation.
Condensation
Moisture which condenses can cause a problem in flues that remain in use. The burning fuel produces water vapour. If the flue is very tall, wide or particularly cold, the flue gases may cool to the point where the moisture vapour condenses within the flue. This happens particularly when wet fuels such as freshly cut timber are used.
Condensation-related damp can usually be reduced by introducing a flue liner, as this both adds some insulation to the flue and reduces its surface area.
Ingress of rain
There are three likely routes by which rain can enter the structure: simply down the flue and into the building; around defective flashings between the chimney and the roof; or through the wall of the chimney stack itself where the fabric is too thin or too porous to prevent penetrating rain from getting around the flashings.
Rain can usually be prevented from coming down the flue by introducing a fairly discreet capping. Types are available for flues which are no longer in use (providing ventilation only) and for flues still in use. If it is still in use, the draught may be affected by the capping, causing the fire to smoke, so some experimentation may be required. When a flue has been relined, rainwater which had previously been soaked up by the old parging may run down the new flue as if it were a drainpipe. Rainwater in the fireplace may be a problem after a flue has been relined.
Where flashings are defective, these should be thoroughly inspected and repaired in accordance with usual good practice and the recommendations of the Lead Sheet Association. However, all too often inspection reveals that penetrating rain is getting around an otherwise perfectly sound flashing. Such problems are more difficult to solve, and a degree of intuitive judgement must be used. The BRE and the Lead Sheet Association now recommend taking a sheet of lead all the way through the structure of a chimney stack just above the roof-line, but this will not usually be an appropriate solution in conservation work. However, the introduction of higher (taller) flashings and ensuring that all pointing is sound may help.