If travelling with luggage in SE Asia and not staying in hotels, where can I leave my belongings?

Yes and no. But let's back up a bit and establish a few things first:

• No one can make blanket statements about entire classes of accommodations; there are a huge variety of hostels, for example, and the amenities will vary widely from locale to locale and even within the same locale. You should always confirm what amenities and facilities are available at each place— and what they will cost, as hostels and the like are no better than hotels when it comes to nickel-and-diming the guests.
• As you acknowledge, it behooves the traveler to travel light, and to take appropriate gear. Rolling luggage is often impractical for the backpacker/flashpacker; even if you are just traveling from one urban center in Europe to another, and can expect sidewalks to exist, there are many places where staircases and curbs are the norm, and elevators/ramps a rarity (e.g. Metro stations, cobblestoned town centers). I find this a pain in the neck even transferring flights within some airports, and so almost always use a duffel and/or backpack instead of a roll-aboard. I rent all my scuba and ski gear; it hasn't ever been worth the trouble for me to bring it from home.
• You don't necessarily have to stay at a hotel to stow your bags there; at least, I have never had a problem checking a bag with the bell desk at a large hotel in the U.S. Having a place to stash my things would not be a reason to choose to stay at a hotel.

It is not at all unusual for travelers to check out of their lodgings in the morning, but have no onward travel until later in the day. Almost every place you stay at will have some kind of solution for what to do with your bags after checking out. But there are many different "solutions" you may find on offer:

• A large hostel with scores of beds may well operate something like a bell desk where you can check belongings securely, often for a fee.
• In a small- or medium-sized hostel, there is usually just a closet or storeroom adjacent to a common area where departing guests can stash their things for the day. In my experience, access to the area is controlled (e.g. behind a door where only people staying at the hostel are allowed to go), but the room is not actively monitored and probably not locked. Everyone's in and out all day, finding a spot for their things, then retrieving them later. This has obvious risks, but in over a decade of travel I can't say I've ever had anyone take an interest in my dirty laundry.
• Usually, you can check valuables like a computer or camera with the front desk staff, who will put it into "safekeeping." Depending on the facility, safekeeping may mean an industrial safe, a cabinet in a private office, or simply the bottom drawer at the front desk. As always, use your judgment. When the the hostel is family-owned and it's the matriarch at the front desk, I might prefer the drawer at her side to the safe in the back where her grandson's sketchy friends hang out all day.
• Some places have lockers in common areas which you can rent, either through the staff or as a coin-operated amenity. Some lockers may be large enough to accommodate a large backpack, while others may barely fit a a phone and wallet. At once place in Bangkok, the lockers were free but you had to provide your own lock (and open lockers were hard to come by, being first-come first-serve).

Depending on where and when you're traveling, there are cases where it might be easier or safer to store your luggage elsewhere.

• As noted above, the bell desk at a full service hotel will be happy to keep your bags for you. Getting the night staff at a no-tell motel to do so will more challenging.
• If you have a rental car, you can keep things in the trunk, but be forewarned that in certain locales (Hawaii in particular comes to mind), rental cars are heavily targeted by thieves.
• In many European and East Asian countries, you can find coin- or card-operated public lockers at bus and train stations. The Cologne Hauptbahnhof had a nifty automated system that takes your bag away by conveyor belt; no risk of vagrants jimmying loose doors off of the lockers (admittedly, not a big risk in Cologne regardless; Nairobi on the other hand...).
• Larger train stations and airports may have a left luggage service that will store your bags securely for a fee. In the U.S., this is virtually the only option remaining at those facilities as all self-service lockers have been removed since September 11.
• In some situations, it might be worthwhile you can check your bags in early for your onward connection. Airlines will check in bags at least three or four hours before your flight— if they have staff around to check you in of course. I had an 0730 LHR-BRU flight on BMI back in the day where the staff didn't show up until 0615.