Chopped bark looks nice and organic but is not normally a good choice as bunny bedding. While it’s safe for rabbits to walk upon in that it won’t hurt their paws, it isn’t at all absorbent and therefore is useless for a litter tray. More importantly, certain types of bark, including those some of those sold as mulch for gardens, are highly toxic. If your rabbits nibble such bark, they could die.
Aside from looking pretty, nontoxic bark is better than a bare wire or plastic floor. Bare wire cuts feet, and rabbits can slip on plastic, hurting themselves in the process. If your rabbits have a large cage, bark could be a possibility for the main run but not for the sleeping or toilet areas. Large chunks of bark provide gnawing opportunities, helping to wear a rabbit’s ever-growing teeth down.
Bark absorbs very little moisture and so is completely useless for the spots your rabbits use as a toilet. In an outside run, its lack of absorbency may actually be an advantage in that it won't get fouled from rain. Grass, however, is better. If you find your rabbits are eating all the grass, exposing bare soil, consider moving the run and making it larger. If your rabbits have free access to part of your garden or you walk them on harnesses, confirm that everything they might encounter, including bark mulches, is safe.
If you don’t know what tree species a type of bark came from, don’t use it. Poisoning can cause paralysis and death in rabbits, even if treated immediately. Milder cases can lead to long-term health problems. Safe tree species for rabbits include willow and certain fruit trees, including apple. Dangerous ones include pine and other conifers, peach, plum and cherry. Anything treated with pesticides or preservatives also poses a hazard. If in doubt about a bark, consult your vet or skip the bark plan altogether.
Straw or hay is probably the best option for the sleeping area at least, being warm and comfortable. Since rabbits should have an unlimited supply of hay to eat anyway, you might as well use some of it for bedding. Hay isn’t especially absorbent though, so use a paper-based substrate for the litter trays. Newspaper works well for this purpose, although don’t use it elsewhere in the cage -- the ink, although not very toxic, stains fur. A commercial paper or cardboard bedding can be used throughout.