This is best done on a sunny day. Lloyd Loom pieces are made from twisted paper and wire and have to be treated with care.
Vacuum with the nozzle attachment and use a dry, stiffish brush (an old scrubbing brush is good) to loosen the dust. Then wash down with a weak solution of washing-up liquid and warm water, paying attention to any very dirty areas.
Use a softer brush to help dislodge the dirt in between the weave, but take great care not to over-wet.
Rinse and allow to air-dry outside in the sun.