In fact, using a cloth to wipe clean the bike dirt are quite eco-friendly. You might use less water if doing it properly. It also reduce the risk of splashing water into bearing,etc.
For heavy dirt like mud, first use gardening spray lightly spray to wet up dirt in order to remove them. Just be careful when dumping the waste water, toilet bowl is a good idea compare to bathtub drain.
For greasy part, spray grease removal/solvent and use paper towel instead of cloth. Don't feel bad about "saving the tree" on the paper towel. Though cloth are "reusable" ,you will waste water and pouring those greasy dirt into your drainage, which is more harmful to the environment.