On schedules
I tried the schedule thing, but it wasn't for me and can't really recommend it. Can't keep up the discipline to myself to do it every time I scheduled any chore reoccuring weekly or monthly. I would find giving myself free passes from a chore, simply when another more important thing comes up or whatever. Well it worked for a few weeks, but later on, it doesn't get done at all. Guess from that perspective it is a time management and priority/objective thing for me that I have to sort out on a day by day basis (how badly do you want to do this and this today?). But yeah, if you can make it work from a schedule for yourself, why not?
Another way of time management
What works for me is a priority list. I made and maintain a list of priorities of items I find important and keep it with me at all times. The items slowly shift priority and new items get added and other things of very low priority sometimes go off. Among other stuffs, my priority list will include things like: "having clean teeth", "having cool hair cut", "having enough groceries", "having a clean bed", "having clean clothes ready to wear", "having a clean toilet", "having a clean house", "having a clean e-mail box", "having work done on my hobby", all the way down to entertainment (which happen to be of lowest priority to me). Then whenever I want to plan for the upcoming day, or whenever I'm like "ok, what to do next?", i open the list on my phone. start on the first item and ask myself if requirements for that item are met or not. If not, get to it or schedule it for the upcoming day. So with "having cool hair cut", i'd check the mirror, apply gel if needed and decide if I make appointment with the barber right away or not. Cleaning house is half way down the list and every time I get to that item I'd check my place and see what needs cleaning most, pick a single thing and dedicate myself to that. Note that the toilet is always a loose thing on my list that gets special attention in my case. I just find it fucking important that the toilet doesn't become smelly and stuff. It is the one place that also gets special exception on the rule down below to not do things room by room.
No room by room multitasking - Work in efficient chunks
When choosing a cleaning activity (or any activity for that matter) to do, I found it to be more effective and time saving, doing the same thing in a single go. In this case doing one cleaning action for the whole house, instead of doing multiple things room by room. So, if the floor wasn't nice in some rooms, I'd vacuum and mop the whole place, every room, given my floor is all tiles everywhere. Otherwise, if you have floors that require special treatment, you might want to split this up in separate chunks maybe. If there is dust on the table in the living room, i'd dust every table, desk or whatever surface around the house. If a mirror doesn't look nice, I'd clean all the mirrors and I'd include the shower cabin, as I kinda use the same stuffs there. If the oven is dirty, i'd clean all the kitchen appliances I used. In the end it would save me time not having to switch cleaning equipment and trying to multi task my way around a single room. You also tend to get faster at something you do, after repeating that certain action over and over for a longer period of time. A bit like getting in a flow, I guess. The only exception on all this for me being the toilet, as pointed out earlier... I'd almost daily ask myself: is the toilet meh? If I think it is, I'd clean the whole damn thing. Ending up doing it about twice a week on average.
Let someone else do it
All the above I did for over a year alone, but now I let my girl take care of most of the cleaning and laundry, so I can spend my time better more productively which will in the end be better for me / for us. Hiring someone also comes to mind, if you can earn back that investment with the time saved by not cleaning, is also a way to be more efficient at cleaning, isn't it?.. That said, I still have this item on my priority list and whenever it pops up I ask myself what needs attention and then ask myself if I want / can ask my girl to do it or not. At times I still end up cleaning the toilet or mopping the floor, just because I really want too and my girl is busy as well (with ironing for example). Just the way things go. How badly do you want it? You want it badly enough? Then do it.
To wrap up
In the end I only clean when I really need to clean something and I also only clean after things of more importance to me are cleared from my list first... I think that's as efficient as it gets from a time management perspective. You don't waste hours on it if you don't need to, but you invest the time needed, because in the end it is important.