Consider the problem of Private Information Retrieval (PIR) where a user wishes to retrieve a single message from N non-communicating and non-colluding databases (servers). All servers store the same set of M messages and they respond to the user through a block fading Gaussian Multiple Access Channel (MAC). The goal in this setting is to keep the index of the required message private from the servers while minimizing the overall communication overhead.This work provides joint privacy-channel coding retrieval schemes for the AWGN MAC with and without fading. The schemes exploit the linearity of the channel while using the Compute and Forward (CF) coding scheme. Consequently, singleuser encoding and decoding are performed to retrieve the private message. The achievable retrieval rates are shown to outperform a separation-based scheme for which the retrieval and the channel coding are designed separately. Moreover, these rates are asymptotically optimal as the SNR grows and are up to a constant gap of 2 bits per channel use for every SNR.