My horse was green broke when I got her and wouldn't pick hers up either. I would always do the hooves in the same order (left front-left rear-right rear-right front), just to try to sort of teach her what was coming next so she could prepare herself.
When I went to pick the first one up, I would run my hand down the back of her leg and pinch when I got towards the bottom with my fingers - sometimes that encourages them to lift it. With my horse, just doing that wasn't enough...so I had to pinch, then basically lean into her shoulder with my shoulder to try to off balance her and get her to lift it. Once she lifted it, I'd pretty much have to GRAB it and hold, while she more or less wobbled around, and pick it as quickly as I could. Then repeat.
My horse has very little muscle tone when I got her though so it was likely difficult for her to pick up her feet. After awhile when she would use me as a crutch and lean back onto me, I would step away slowly and hold the foot, she always fell over once, but caught herself at the last minute. Once she figured out that I wasn't going to hold her up, she quit trying that.
After awhile of building up her muscle and strength, she didn't do it anymore at all and eventually she got to the point where now she anticipates the next hoof I'm going to pick up, and rests it up for me before I even get to it