  In this paper we focus on the almost-diagonalization properties of
$\tau$-pseudodifferential operators using techniques from time-frequency
analysis. Our function spaces are modulation spaces and the special class of
Wiener amalgam spaces arising by considering the action of the Fourier
transform of modulation spaces. A particular example is provided by the
Sj\"ostrand class, for which Gr\"ochenig exhibited the almost diagonalization
of Weyl operators. We shall show that such result can be extended to any
$\tau$-pseudodifferential operator, for $\tau \in [0,1]$, also with symbol in
weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra
and Wiener properties for $\tau$-pseudodifferential operators on Wiener amalgam
and modulation spaces.
