How old are your old big appliances? How many of them?
Mine appliances are 50-75 years old (big stainless steel old fridges, old water tanks, old ice cream makers, old milk fridges, old stoves, etc) and it took more than three median sized trucks. If I do not want to sell the house, I will leave them there.
If your stuff is not that big and heavy, you can ask any handyman who has a pickup truck. It is only about $50 for a pick up truck load.
It really depends on what you have and find an economic way to get rid of it.