From what I know, there are two ways people usually let go. One, set the intention, feel it as if its a reality now, and totally forget about it. Just distract yourself by involving in some other tasks and get on with your life happily. This is the method that works the best for me. When I set an intention and forget it, it manifests really quick. Sometimes just within hours after desiring! What is happening here is when I totally forget about the desire I am no longer blocking the path of allowing my desire into my life. I have no attachment about it because I am not even thinking of it. Since I have already set the intention and felt it into being I do not have to do it again and again. The Universal forces get into motion the moment the intention is set. So by forgetting, I totally let go of all the attachment with the desire and it manifests in no time. So my motto is 'Set it and Forget it'
Two, as I already mentioned letting go is not absence of desire itself, its the absence of negativity associated with the desire. So it is, in essence, letting go of negative feelings about the desire. So many people find that when they think of something they desire with absolute faith and loving thoughts, without a grain of doubt, and they feel happy even without the desire, they tend to manifest their desires easily. It is because they have absolute faith therefore they have let go of any worry/doubt etc. Also they are happy even without the desire, the desire is just an icing on the cake. In such a vibration, they are not needy or clingy towards their desire. They have let go of the attachment. Hence things manifest easily and really fast