What are your most effective ways to motivate the unmotivated?

Motivate by example. Everyone has a bad day, but you can't reflect it and never share the grief with others. It can be contagious and really damage morale. If you set an example that others like, they will follow. Peer pressure is a great tool to bring the laggards aboard with the rest. It's all basic team building. If people prefer to work with you rather than for you, and you reciprocate, you only have to deal with the 10% (time bandits). They might even fall in line, or they fall out.

I learned early on that if I tap into what interests them, will also motivate them. Find that way to connect being a soldier to what motivates, they will come around. It also shows that someone cares about them. For a young person set out into this world, knowing someone cares is important to them.
I had a young sailor on my 2nd boat that would not respond to any disciplinary action and was soon to be kicked off the boat. In submarines, if you do not maintain your qualification schedule, and fall way behind, they will remove you from the boat. He was almost there. I took him under my wing and found his interest. It happened to be art. My collateral duty was the Chair of the Recreation Committee and we did a little news letter while on patrol. I had him contribute a drawing to the news letter. He gained a little notoriety and responded by getting hot on his quals. Last I heard, he retired as a Master Chief.

Unmotivated people cannot connect the dots to see the rewards of good work, diligence, etc., and frequently don't care. The problem is we spend more time than we should on most getting a result. When I was skippering, my policy was to tell the XO and SEL that we're making business decisions, i.e. is the investment worth it? Will improvement to this level make the SM competitive for promotion or are we kicking the can down the road to the same result. Unmotivated people are a albatross around the command's neck and a morale killer. Deal with it as such. My first command, I had 3 SMs gone within two months and by month 6, a few more. It was shock treatment that actually helped some to connect the dots. Funny thing, your reputation leads ahead of you. Prior to my next skipper job, the skipper I was relieving started cleaning house before I showed up. I will admit that I was very much in the minority, but I was heart attack serious about good people not being dragged down. The NCO community was appreciative of my approach. That said, nobody was exempt. I canned an O-5 Ops Boss and an E-8 as well.