Firstly please, please don't try to vacuum them. It may seem like a smart, quick way to remove dust but in reality a 'good idea' can turn into a complete disaster with the slip of the hand. You don't want to damage your piece so best to stay away from high powered equipment altogether. Instead if you find your piece is starting to look a little yucky take it outside, hold it up by the top (the dowel, branch, ring, whatever it's mounted to) and gently shake it from side to side. This will dislodge any dust particles and also help move all the tassels and fringes into place. If you find that there is still some gunk hanging on, you can take one of those sticky lint rollers and carefully clean the stubborn areas. You can also use the back of some low tack tape to the same effect.
If you accidentally get some staining on your piece I would advise against submerging it in any liquid. Some of the materials used to make it may shrink and cause the design to become distorted. Spot cleaning with some lukewarm water or diluted white vinegar can help remove any spots but make sure you spot test on the back or an inconspicuous area first to see how the materials react.
The disinfecting properties of sunlight can help to remove any musty smells that may affect older pieces. Make sure it's not likely to rain or become too windy and keep a close eye on your piece throughout the day. I would advise against leaving it out for any longer than a few hours.
Keeping your hanging away from moisture and cooking fumes is a must. Moisture can cause it to become smelly and can encourage mould growth. Mould will permanently damage your piece.
Some natural fibres can attract bugs. I have never seen or heard of this happening but better to be safe than sorry. Periodically check your piece for any signs of creepy crawly activity.
Regularly checking your piece for signs of dust, bugs, mould and stains will help prevent any little problems turning into art destroying nightmares leaving you to enjoy your fibre art for many years.