We explore approximation algorithms for the $d$-dimensional geometric bin packing problem ($d$BP). Caprara (MOR 2008) gave a harmonic-based algorithm for $d$BP having an asymptotic approximation ratio (AAR) of $T_{\infty}^{d-1}$ (where $T_{\infty} \approx 1.691$). However, their algorithm doesn't allow items to be rotated. This is in contrast to some common applications of $d$BP, like packing boxes into shipping containers. We give approximation algorithms for $d$BP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm having AAR $T_{\infty}^{d}$. We next give a more sophisticated harmonic-based algorithm, which we call $\mathtt{HGaP}_k$, having AAR $T_{\infty}^{d-1}(1+ε)$. This gives an AAR of roughly $2.860 + ε$ for 3BP with rotations, which improves upon the best-known AAR of $4.5$. In addition, we study the multiple-choice bin packing problem that generalizes the rotational case. Here we are given $n$ sets of $d$-dimensional cuboidal items and we have to choose exactly one item from each set and then pack the chosen items. Our algorithms also work for the multiple-choice bin packing problem. We also give fast and simple approximation algorithms for the multiple-choice versions of $d$D strip packing and $d$D geometric knapsack.