A linear LED lamp is disclosed. Embodiments of the invention can provide an LED-based replacement lamp for a linear or “tube-type” bulb or a bulb with a linear filament or element. By filling the void within the lamp with an optically transmissive fluid to cool the LEDs without the use of a traditional heat sink, the light blocking effects of such a heat sink can be avoided. Thus, the LED replacement lamp can emit light in a substantially omnidirectional pattern. In some embodiments, the optically transmissive fluid medium is a liquid. In some embodiments, the optically transmissive fluid medium is a gel. An index matching medium can be used as the optically transmissive fluid medium. A color mixing treatment can optionally be included to eliminate color tints in cases where multiple LEDs of different colors are used to produce white light.