Two major approaches are being followed: atom counting and electrical approaches. The atom counting approach is easy enough to understand-count every atom in a sample then multiply by the mass of the atom and then you have the mass of the entire sample. Unfortunately we can't do this yet. What we can do is grow very pure crystals. The leader in the atom counting field is the Avogadro Project. They are seeking to make a single crystal silicon sphere, using laser interferometry to measure the diameter and then because they know the crystal structure of silicon they can calculate the number of atoms in that sphere.
Why use silicon? Because technology has been developed to purify silicon to a higher purity than anything we can make. Silicon used in the semiconductor industry is >99.99999% pure. It is made through a multi-step process first with zone refining and then a single crystal boule is grown from pure molten silicon. The isotopic make-up is then measured with a mass spectrometer and average atomic mass is then calculated. This single cryastal is then cut and polished into a perfect sphere. Alright it isn't perfect but it is nigh-perfect. To quote Wikipedia:
On the Ø 93.6 mm sphere, an out-of-roundness of 35 nm (undulations of ±17.5 nm) is a fractional roundness (∆r/r) = 3.7 × 10–7. Scaled to the size of Earth, this is equivalent to a maximum deviation from sea level of only 2.4 meters. The roundness of that ACPO sphere is exceeded only by two of the four fused-quartz gyroscope rotors flown on Gravity Probe B, which were manufactured in the late 1990s and given their final figure at the W.W. Hansen Experimental Physics Lab at Stanford University.
The problems that they are facing are things like too much variation in the isotopes of the silicon and the thickness of the silicon oxide layer that grows on the sphere.
The second proposed method for redefining the kilogram is called the electronic kilogram. The idea is to design an incredibly sensitive balance that matches the weight of an object (the force exerted on an object by gravity) to an electromagnetic force produced by a coil of current carrying wire in a strong magnetic field. Then you could define the kg as 'the mass that can be suspended by the electromagnetic force generated when a specific amount of current flows.' So far the electronic kilogram has produced better results than the atom counting method, but it still can't provide an accuracy of a millionth of one percent every time.
1. Robinson, Andrew. The Story of Measurement. London: Thames & Hudson, 2007.