I NEVER USE BLEACH. I also use a steamer to clean their cages. I do it once every couple of weeks. But I do a wipe down every day. Bleach will leave a residue no matter how well you clean it off.
Bleach fumes are harmful to birds and bleach should never be used while a bird is in the cage (or even the same room, or house!)
However, it is not true that bleach leaves a residue when dry. When bleach dries (pure bleach) it crystallises through a chemical reaction to plain old salt. Provided you are using pure bleach (no additives like lemon scent or whatever), hose it down and let it dry, any residue will be plain old salt. NaCl. What you put on you food!
I work in the aquarium industry. Koi in particular. Sanitising empty aquariums, ponds and holding tanks with bleach is standard procedure as because once it has dried out it is non toxic, as it is salt. We use a 1:10 ratio of bleach to water, and that is considered very high. I:25 would be more appropriate IMHO. So if we miss some in the final rinse (after drying) and fill the tanks back up, the worse thing that can happen is a tiny bit of salt in the water. If bleach left a toxic residue, it would kill those (very expensive!) fish almost overnight, as has happened with some other 'safe' cleaning products (Including GSE). Therefore, we use bleach for safety reasons, with a procedure that involves a 24 hour drying time (overkill, but for safety) for the koi's safety. We also have in the past used PP and GSE to satinise aquariums and holding tanks, both of which left more toxic residues (to the fish, so this does
not translate to bad for your bird!) then the trace amount of salt that pure bleach, when used correctly, would.
Therefore, if you are taking your cage outside, without the bird in it to clean it, it is safe to use bleach provided you rinse it well and let it dry 100% so any residue crystallises to salt. I would not do this as a regular cleaning procedure, but if you just bought a new or secondhand cage and needed to satinise it, or you have had a ill bird in a cage and need to sterilise it, it is one of the surefire ways to actually, safely, do so.
As for cleaning my cage? I wipe the grate down and change the paper daily, as well as any obviously dirty spots. I also clean all bowls daily before filling them (twice for the fruit one, cleaned every time it gets filled, and sometimes more then once for the water bowl) Weekly, or as needed, I take Alex out and give the cage a good clean with vinegar with a drop or two of eucalyptus oil mixed in from a spray bottle and wipe it down. It is a bird safe antifungal and antibacterial mix. I use the same 'product' (home made really) around the house actually. One or two drops of oil is sufficient, and you should not be able to smell it after an hour once it dissipates! Once every few months I also take the cage outside and give it a 100% scrub down with whatever is handy and safe. Usually bleach or a bird safe sanitiser (Reptile aisle at your local pet store will have plenty).