What method to use when cleaning an old wall-mounted garage heater?
A good way to get started with cleaning it would be these steps:


Remove the whole array of front panel grills. It may be a single grill or multiple. 
Carry grills outside and scrub with warm soapy water. A scrub brush will work well, especially for the grill slots.  A wash cloth may work best on the large flat areas.
Spray down the grill(s) using garden hose to flush away the soap and loosened dirt. Set grill aside to fully dry.
Use a vacuum cleaner (shopvac if available) to clean up the dust, dirt, pet hair, and other debris in the furnace enclosure itself. A brush attachment on the end of the vacuum cleaner hose can help a lot on the irregular surfaces. Once the major stuff is cleaned then a damp wash cloth can be used to wipe down the inner surfaces of the furnace.