As in the comments and earlier answer: in short, there is nothing comparably elementary or accessible for GL(3), as holomorphic things for GL(2).
Even the explication of this apparent fact is not, and perhaps could not be, as immediate as the direct exhibiting of holomorphic things for GL(2): to demonstrate the absence of things is harder than showing presence.
The most "elementary" GL(3) forms might be the Gelbart-Jacquet lifts from GL(2), although what is readily describable is not the automorphic form(s) but the L-functions. In any case, this is a "global" description which, therefore, is inevitably more complicated.
A "local" description of (unitary?) repns of the GL(3,R) versus GL(2,R) is much easier than global comparisons. GL(2,R) basically has holomorphic discrete series and principal series, the former (globally) giving holomorphic automorphic forms, the latter giving waveforms. The repn theory of GL(3,R) includes no discrete series (this is not obvious). It does include both (irreducible) principal series and repns induced from the holomorphic discrete series on the copy of GL(2) in the 2,1-block or 1,2-block parabolic subgroup(s). In the last few decades, the "cohomological" repns have been distinguished, although this is less easy to describe than principal series or induced repns generally.
In practice, none of these repn types are as accessible as the holomorphic things for GL(2) on the upper half-plane.
The older thread in which holomorphic automorphic forms were the exclusive topic was Siegel modular forms, indeed. To compare the repn theory, this was possible because Sp(n,R) (or "2n"...) does have holomorphic discrete series repns, for all n. Siegel and H. Braun did not address things in such terms, and Harish-Chandra's early work on repn theory seems to have overlooked the implicit appearance of holomorphic discrete series in Siegel's work!
Sp(n,R) (2n-by-2n matrices) in fact has $n$ different discrete series. For example, the 4-by-4 case has the holomorphic ones, and what are called the "big" discrete series. The repn theory also includes all induced repns from Levi components of parabolics. Thus, in fact, the relative poverty of unitary repns of GL(n,R)'s gives them a greater simplicity than the Sp(n,R)'s, despite the fact that the story is not elementary.
Certainly holomorphic discrete series are among cohomological ones, and I believe (Clozel more-or-less confirmed this a few years ago) that at least case-by-case Vogan and his students have shown that all discrete series are cohomological.
It is also considerably surprising that the old (1960s) results of Matsushima and Murakami on which automorphic forms can appear in cohomology really does show that it depends only on the archimedean representation type.
So we probably must reconcile ourselves to "cohomological" being the "right" generalization of "holomorphic".