Holographic multiple-input multiple-output (MIMO) systems constitute a promising technology in support of next-generation wireless communications, thus paving the way for a smart programmable radio environment. However, despite its significant potential, further fundamental issues remain to be addressed, such as the acquisition of accurate channel information. Indeed, the conventional angular-domain channel representation is no longer adequate for characterizing the sparsity inherent in holographic MIMO channels. To fill this knowledge gap, in this article, we conceive a decomposition and reconstruction (DeRe)-based framework for facilitating the estimation of sparse channels in holographic MIMOs. In particular, the channel parameters involved in the steering vector, namely the azimuth and elevation angles plus the distance (AED), are decomposed for independently constructing their own covariance matrices. Then, the acquisition of each parameter can be formulated as a compressive sensing (CS) problem by harnessing the covariance matrix associated with each individual parameter. We demonstrate that our solution exhibits an improved performance and imposes a reduced pilot overhead, despite its reduced complexity. Finally, promising open research topics are highlighted to bridge the gap between the theory and the practical employment of holographic MIMO schemes.