It should be fine as long as the plastic containers or bags have lids and the bags are tied. I imagine that as long as there are no loose objects that can fly. Local moving companies may agree to carry some of your things in garbage bags, as the distance is shorter and the less time you'll spend on the road. As long as the soft, non-fragile items are packed inside the plastic bags, the moving company may agree.
However, moving companies across the country are likely to refuse to take items packed in plastic bags due to the longer distance (hundreds or thousands of miles) and the long delivery time. The risks are too great, so long-distance movers want everything neatly packed in sturdy cardboard boxes for added protection. In any case, when it comes to packing, garbage bags are ideal for packing fabrics. Since items such as clothes or bedding cannot be broken, you can effectively pack them in a garbage bag.
Of course, you'd like to use stronger packaging materials if you've hired storage services and plan to leave your closet for a while. Otherwise, fold the clothes or bedding neatly and place them in the bag. This is one of the best ways to use trash bags when you're moving. The fastest way to pack is to start early, do focused sprints, and use a methodical room-by-room strategy.
Don't try to multitask in multiple rooms, pack efficiently by running out one space before moving on to the next. Use trash bags to store plush items, closet boxes to hang clothes, and keep your packaging items close at all times so you don't waste time looking for ribbons or labels. Speed comes from preparation, not from haste. If you're moving in the winter, plastic trash bags can also protect your hardwood or carpeted floors from snow.
The first time I moved on my own, I hired a moving company to come and get some of the larger furniture, but I also have some boxes and some items that don't fit in my car. Sturdy garbage bags are ideal for packing some items, but you'll have to ask the moving company what they can and can't be used for. Some items fall into a gray area, meaning that some moving companies will fill them out, while others will require them to be packed. In some cases, if you move to a location, these items can be left unpacked, but the moving company may ask you to sign an exemption of responsibility.
Moving companies take almost everything, but there are some items that you will have to discard, give away or move in yourself. Either way, before you stock up, ask the full-service moving company if they have free or discounted used boxes. Save for last the areas of greatest use, such as the kitchen or bedroom, since you may need access to those items before the moving companies arrive. This step saves time when loading the moving truck and helps moving professionals get things back together quickly at your destination.
Using garbage bags during a move is definitely an interesting idea that has the potential to save you time and money during the house packing process. However, the real question is: will moving companies move garbage bags? When you use professional moving services.