We present InvMSAFold, an inverse folding method for generating protein sequences optimized for diversity and speed. For a given structure, InvMSAFold generates the parameters of a pairwise probability distribution over the space of sequences, capturing the amino acid covariances observed in Multiple Sequence Alignments (MSA) of homologous proteins. This allows for the efficient generation of highly diverse protein sequences while preserving structural and functional integrity.We demonstrate that this increased diversity in sampled sequences translates into greater variability in biochemical properties, highlighting the exciting potential of our method for applications such as protein design. The orders of magnitude improvement in sampling speed compared to existing methods unlocks new possibilities for high-throughput in virtual screening.