The Klein-Gordon equation is solved approximately for the three-dimensional harmonic-oscillator potential. Mathematically, the radial equation has the form of the nonrelativistic equation with anr4anharmonic term added. We carry the calculation to lowest relativistic order and get an energy spectrum with the following, intuitively expected, features: the level spacing decreases with increasing energy and theldegeneracy is broken. The same result is also obtained by putting the relativistic kinetic energy into the Schrödinger equation and expanding inp2.