What organizational methods pre-date the Periodic Table?
One attempt to order chemical elements was Döbereiner's system of triades, published in Annalen der Physik und Chemie, back in 1829 (doi 10.1002/andp.18290910217 with Wiley); or (open access with Gallica).

Although an actual view into his paper permits the speculation he was not using the atom masses we know today; rather than using specific weight and molecular weight as known / defined at his time of oxides of Ca, Ba, Sr, for example:



(The dots on top of Ca, Ba, Sr seem to adhere a nomenclature, as on top of S, Se, Te three of them; Cl, Br, I are topped by five.)