Luneburg lenses have superior performance compared with conventional lenses made of uniform materials with specially designed surfaces, but they are restricted by the difficulty of manufacturing the required gradient-index materials and their spherical focal surfaces. Recently, a new two-dimensional (2D) imaging lens was proposed and realized using transformation optics. Such a 2D lens overcomes the aberration problem, has a flattened focal surface and is valid for extremely large viewing angles. Here, we show the design, realization and measurement of a three-dimensional (3D) approximate transformation-optics lens in the microwave frequency band. The 3D lens is made of non-resonant metamaterials, which are fabricated with multilayered dielectric plates by drilling inhomogeneous holes. Simulation and experimental results demonstrate excellent performance of the 3D lens for different polarizations over a broad frequency band from 12.4 to 18 GHz. It can also be used as a high-gain antenna to radiate or receive narrow beams in large scanning angles.