The technological growth combined with the exponential increase of wireless traffic are pushing the wireless community to investigate solutions to maximally exploit the available spectrum. Among the proposed solutions, the operation of Long Term Evolution (LTE) in the unlicensed spectrum (LTE-U) has attracted significant attention. Recently, the 3rd Generation Partnership Project announced specifications that allow LTE to transmit in the unlicensed spectrum using a Listen Before Talk (LBT) procedure, respecting this way the regulator requirements worldwide. However, the proposed standards may cause coexistence issues between LTE and legacy Wi-Fi networks. In this article, it is discussed that a fair coexistence mechanism is needed to guarantee equal channel access opportunities for the co-located networks in a technology-agnostic way, taking into account potential traffic requirements. In order to enable harmonious coexistence and fair spectrum sharing among LTE-U and Wi-Fi, an adaptive LTE-U LBT scheme is presented. This scheme uses a variable LTE transmission opportunity (TXOP) followed by a variable muting period. This way, co-located Wi-Fi networks can exploit the muting period to gain access to the wireless medium. The scheme is studied and evaluated in different compelling scenarios using a simulation platform. The results show that by configuring the LTE-U with the appropriate TXOP and muting period values, the proposed scheme can significantly improve the coexistence among LTE-U and Wi-Fi in a fair manner. Finally, a preliminary algorithm is proposed on how the optimal configuration parameters can be selected towards harmonious and fair coexistence.