I grabbed my chatty responses to her and put them together in this brief Crystal Cleansing Guide for you. I also included some resources at the bottom for further reading.
Moonlight and Sun
This method is much talked about during the times of the Full and New Moon. It is simply setting your crystals outside or in a window to be bathed in the moonlight, cleansed, and empowered. The same is done with sunlight for similar reasons.
You can get really complicated with this such as using astrology to empower crystals with certain signs or use only certain stones based on their properties (feminine, water element, deity connection, etc) for moonlight cleanse and certain others (masculine, fire element, etc) in the sun. A good, easy example of the last is moonstone in moonlight, sunstone in sunlight.
Colorful crystals such as amethyst, citrine, or fluorite can suffer from sun-bleaching which will fade and drain their color. This does not hurt the crystal or make its vibration weak, I understand, but it might not be as pretty or vibrant after a few days in direct sunlight.
I say go with whatever floats you're boat and feels right on this one.
As for moon phases that are best for cleansing, I always following the idea that Waning (decreasing) and New Moon are best for cleanse and renewal and Waxing (growing) and Full Moon are best for empowering.
Spring water, ocean water, holy water, moon water, even tap water has been used to cleanse stones.
I personally don't like or recommend tap water as its energy is not as clean to me, but that is my opinion. The idea behind it is the water flows and takes the non-beneficial energy down the drain and, presumably, back to the earth. You can use this concept by cleansing your stones in a stream or by running water over them outside by pouring your chosen libation over the stones and letting the residual fall to the earth below.
For cleansing stones I use moon water more than any other. Moon water is made by placing a glass jar or bottle of water in the moonlight to collect its energies and then storing it in a dark place until needed. I usually do this during eclipses, super moons, and other rarer occasions. When I want to cleanse or empower my crystals, I don't have to wait for a moon phase. I simply grab my moon water of choice and give them a bath (I do this with myself and other tools that can handle water as well).
Some stones that are naturally soft can dissolve in water - example: Selenite. Others, like hematite, will rust when in contact with moisture. Check on your stone type and the effects of water on it before bathing...sort of like a Gremlin, though I've never tried feeding my stones after midnight.
Of course also be wary of washing your crystals or leaving them in water if they are set in jewelry or tools such as pendulums as water will tarnish or rust some metal pieces.
Burying your crystals in earth or in salt is done with the idea in mind that stone come from the earth and by connecting them back to their mother element they are rejuvenated and cleanses.
You can do this outside in the ground (be sure to mark the spot and draw yourself a treasure map) or indoors in a garden pot or in a simple bowl.
Just like with water, some stones are soft and salt can cause them to become brittle and break. Be gentle.
Wafting the perfumed air of a burning smudge stick or incense around a crystal or stone, bathing it in the smoke is a great way to cleanse. People do this to themselves and their space all the time, why not to crystals?
Make sure to use a smudge bundle or incense that is appealing to you - it would do no good to your crystals if you're choking on a noxious scent the whole time you're trying to do a cleansing. That would be like trying to clean your home with cleaners and room sprays that give you headaches and make you sick, who wants to be in that home afterwards?
Singing bowls, bells, mantras, even a CD of singing monks can cleanse stones. Done in a similar manner to smudging, the sound waves waft around the item or person being cleansed, shaking off the non-beneficial energies. This method is best for those who, of course, enjoy the sounds of bells and bowls and monks. For some, the bells and bowls especially, are a bit high pitched and there's no need to make yourself or your loved ones uncomfortable for the sake of cleansing your rocks. Try another method if that is the case.
This might sound weird but holding your crystals, sending them love and peace, and doing a brief visualization of pure white light encompassing them can be just as effective as any of the aforementioned methods. These crystals are your allies and companions. They are there to help you in healing, in meditation, in your personal and spiritual evolution. Of course they would like your love and blessing in return!
You can also combine the above cleansing methods into a ritual of your design. Some decide to use a cleansing method of all 4 elements; Air, Fire, Water, Earth. Others simply do what feels right or experiment with new ways each time.
Many books on the craft and energy healing will have a section in the index about crystals and their metaphysical properties. I suggest starting there if you already have a library started in this genre.
Hibiscus Moon Crystal Academy - Hibiscus Moon has some great e-books, blog posts, and articles, many of which are about cleansing and using her crystals (she apparently gets asked this a lot too).
Katrina Raphaell's Crystal Enlightenment and other volumes on crystal use. These are my favorite crystal books and I highly recommend them for anyone wanting to delve into the magickal world of crystals, healing, and energy work.