Choose Your Mop
String mops aren't a complete no-no, if you have one with microfiber strings that you can thoroughly wring out -- it's safe for your floor -- as long as the floor's finish is in good condition. You can also use a well-wrung sponge mop. Natural-fiber string mops are virtually impossible to wring it out well enough to avoid splashing excess water on your floor though, which can result in discoloration and warping. Steam mops are even worse for your floor -- they force water vapor through the joints and into the vulnerable unfinished wood underneath.
Ready Your Cleaner
A pH-neutral cleaner is best for your floor, and flooring manufacturers generally recommend cleaning with nothing more potent than clear water. Add an ounce of dish detergent to a gallon of cleaning water to handle dirt buildup. If you need to tackle ground-in dirt or white streaks caused by minerals, mix a 1/2 cup of vinegar with the water. Vinegar is acidic, and it can dull the finish, so test it before using it. Even if you determine that it's safe to use, do so sparingly. The best option of all is to mop with a water-free, commercial product intended for cleaning hardwood floors.
The key to cleaning your floor efficiently while avoiding standing water is to wring out your mop thoroughly before each pass, but avoid scrubbing with the mop. Pass it along the wood grain -- to minimize streaking -- and keep it moving. Mop with the grain to prevent the mop from depositing water on any raised edges where it can seep downward. If you use a cleaning solution, make an extra pass with clear water when you're done mopping with your cleaning solution, and then wrap a dry cloth around your mop to dry the floor.
Vacuuming and Dry Mopping
You shouldn't have to damp-mop your floor often. In most cases, all it needs is vacuuming and an occasional pass with a dry microfiber mop to pick up small dust particles that have settled in the gaps and grooves. A flat mop works most effectively for this, but most flat mops have a small cleaning area that quickly gets dirty, so you may prefer to use a microfiber string mop. When vacuuming, always use a cleaning attachment with soft edges to avoid grinding dirt into the finish. If your vacuum has a beater bar, make sure it's off.