An LED lamp or bulb is disclosed that comprises a light source, a heat sink structure and a remote phosphor carrier having at least one conversion material. The phosphor carrier can be remote to the light sources and mounted to the heat sink so that heat from the phosphor carrier spreads into the heat sink. The phosphor carrier can have a three-dimensional shape, and can comprise a thermally conductive transparent material and a phosphor layer, with an LED based light source mounted to the heat sink such that light from the light source passes through the phosphor carrier. At least some of the LED light is converted by the phosphor carrier, with some lamp embodiments emitting a white light combination of LED and phosphor light. The phosphors in the phosphor carriers can be arranged to operate at a lower temperature to thereby operate at greater phosphor conversion efficiency and with reduced heat related damage to the phosphor. The lamps or bulbs can also comprise a diffuser over the phosphor carrier to distribute light and to conceal the phosphor carrier.