A new moon marks the start of a lunar cycle (symbolising the beginning of a new adventure), hence it's beneficial to make wishes/set intentions under a new moon.
So does that mean we can't make wishes at any other time? Of course not. It is only extra powerful to do so at the start of the lunar cycle. Provided VOC Moon periods are avoided (when the moon is not making any major aspects to other planets), it's still fine to make your wishes at any time you wish.
Recently, I've tried the 'Burn a letter to the Universe' method of releasing my wishes to the greater power out there. All you have to do is address the letter to the Universe (Dear Universe), and then pour your heart out and tell the Universe what you want to manifest in your life. I have no idea who started this practice but I'm sure you can google it....I've personally learnt it from my hypnotherapy and family constellation instructor, and I must say, it's rather exhilarating to see my crisp white paper slowly turning into ashes under the scorching flames.....I've read somewhere that tearing up the paper is rather destructive, burning is a better way of releasing the contents to the Universe....but do take care when playing with fire.....:-P
Image courtesy of Simon Howden / FreeDigitalPhotos.net