Good question! This is one of the biggest, and earliest, decisions you should make. You are also quite right that the most common construction systems are brick and blockwork, but there are other options.
Timber frame is quite popular nowadays due to its (off-site manufacture) fast form of construction. It is also a popular method with builders and a cost effective solution. For a rural look it is also worth considering Green Oak Frame for its warm characteristics and homely charm, it is also not as costly as you might think!
With energy efficiency in the forefront of people's minds and strict workmanship requirements 'Modern Methods of Construction' (MMC) should also be considered. Insulated Concrete Formwork (ICF) uses a polystyrene mould which is then filled with concrete, and usually reinforced, on site, is an incredibly fast route to getting the superstructure up — and it provides an impressively efficient wall structure.
Structural Insulated Panels (SIPs) are essentially a sandwich: the filling is a solid thickness insulation, and the bread is made of rigid building boards such as plywood or orientated strand board (OSB). Again the benefits are factory applied insulation and millimetre accurate construction providing well insulated air tight construction.
Straw bale homes are first formed with a timber frame and the walls are then infilled with bales of straw. The walls are stabilized and covered with chicken wire then rendered for durability and aesthetics. One advantage of straw bale homes is that straw is an affordable and renewable resource, and construction is relatively simple. On the other hand more room is required to accommodate thicker walls and the system is not the fastest on the market.
There are many other construction systems on the market but as with all of these, each method of construction has to be weighed up on a project by project basis depending on location, budget, access, and aesthetics. We at Offset Architects would be pleased to meet with you and explain a particular system in more detail and its suitability for your project.