There are several methods described on this website and summarized here.
Disclaimer, I've not tried any of these except using purified water, which does work well.
Use pure water. This method looks to get as much of the air and mineral impurities out of the water as possible before freezing, start with already distilled water. Filtered bottled water will work, or any water purified by reverse osmosis.
Boiling Method (for the sake of completeness, I realize you don't want to boil water in your question).
Boil the water twice. Boiling removes air bubbles from the liquid, allowing the water molecules to stick together even harder in the freezer. After boiling the first time, allow the water to cool. Then boil again. Keep the cooling water covered to prevent any dust from collecting on the surface.
Cover the tray as it cools to keep particulate out.
Top-Down Freezing Method
Use a cooler to will insulate your ice cubes, forcing them to freeze slowly from the top down.
Place your ice tray, mold or other freezing container at the bottom of the cooler, open to the top. Fill your tray or molds with water.
Pour water into the bottom of the cooler, filling in around your ice tray or molds. This water will seal off your ice cubes, keeping cold air from freezing the sides or bottom.
Place the cooler with the lid off in your freezer. Make sure your freezer is not set too cold—17-25°F should do it. Leave the cooler for 24 hours. away the ice around your tray or molds and remove your ice cubes.
High-Temperature Freezing Method
Set your freezer to just below freezing, around 30°F or -1°C. Fill an ice tray or mold with water and place put it in the freezer. Leave it to freeze for 24 hours. The slow freeze should force out any gasses and impurities, leaving you with perfectly clear ice cubes.
Bottom freezing method
Fill up a bowl with water then dump a lot of salt into it to prevent it from freezing then put it in the freezer. Be careful not to put too little water in the bowl or the freezing will release enough heat to warm up the salt water to 0°C before the ice cubes finish freezing. The colder the freezer is, the higher the concentration of salt needs to be to prevent the salt water from freezing. Leave the salt water in the freezer at least 3 hours to get really cold. Take the bowl of salt water out of the freezer to prevent the water in the ice cube tray from freezing from the top.
Fill up an ice cube tray with water then float it on the salt water in the freezer which is denser than fresh water. The result is bubble free ice that is super strong and crack free because it froze without any trapped regions of water ever forming during the process.
Boiling Method
(added for the sake of completeness, I realize you don't want to boil water in your question).
Boil the water twice. Boiling removes air bubbles from the liquid, allowing the water molecules to stick together even harder in the freezer. After boiling the first time, allow the water to cool. Then boil again. Keep the cooling water covered to prevent any dust from collecting on the surface.
Cover the tray as it cools to keep particulate out.
