Opiates doesn't really take the pain away but makes easier to not be bothered by it so in a combination with other pain reliever it can be effective.
The latest medical opinion is that there are multiple pain pathways, not just one set of nerves. For the majority of patients opiates do "take the pain away", for others, such as fibromyalgia, it does not block it at all.
Without opiates, paracetamol and Non-Steroidal Anti-inflammatory drugs I don't have a lot left to try.
The anti-Inflammatory drugs do work, but by the time they are strong enough to be effective they also start affecting my kidneys. So I can have pain relief - providing that I don't mind going on dialysis permanently in about 6 months.