Yet, the reality can be very different. I know this, because we decided to cloth diaper our youngest kid. Our motivations were being kind to the earth (disposable diapers take around 500 years to disintegrate, which means that every sposie ever made still exists!), saving some money, and convenience. After all, you never run out of cloth diapers. There are actually quite a few different cloth diapering options. Those squares of fabric your grandmother used is one option (with a PUL cover).
Cute diapers that look just like disposables, but are actually made of fabric, are another. We decided to use pocket diapers, which have a stay-dry layer in the inside and a water-resistant fabric on the outside. Inserts that are designed to absorb are placed between those layers, hence the name pocket diaper. All-in-one diapers, which don't require covers or inserts, are another option. And many parents choose to use a diaper service that washes and returns diapers, so the process of laundry is eliminated.
Some cloth diapers come in various sizes, while others come with snaps that allow the diaper to be adjusted in size, often from newborn to potty training. I loved cloth diapers, and would use them again in a heartbeat. After having the experience of using both cloth and sposies, with two different kids, I can honestly say that cloth was easier to use. We washed diapers every two days, and the microfiber inserts we had dried really quickly.
My son potty trained at one and a half, which I think can be attributed to cloth diapers too, at least in part. We never had any problems with diaper rash either, which made a change! One thing to watch out for with cloth diapers is that most baby clothes are designed for use with disposable diapers. Cloth is a bit bulkier, so you will need wider pants.