In the Akkadian, possessive suffixes were frequently used; they were
attached to a noun after the case ending, e.g.
/Sarrum/ "king"
/Sarrumi:/ + poss.suffix "my king"
/Sarri:/   "king's"
/Sarri:ja/ + poss.suffix  "my king's"
For the second and third person, they are differentiated by gender,
i.e. masculine or feminine., and for all persons by number, i.e.
singular or plural. Although they were dual verb forms in Akkadian, I
did not see any hints for a dual suffixes.
The suffixes were quite lasting and can be traced from Old Akkadian
until the end of a living Babylonian language. As Babylonian was used
for certain official purposes even longer, I guess that they can be
found as long as Babylonian was in any use, but I don't know that.

If you wanted to stress the ownership or to express opposites, own
possessive pronouns were available as well. They agree with the noun
in case and in gender. e.g.
  /ma:tam la: kattam/ "this land not owned by you"
land.akk.s.g - negation -
/ma:tam kattam/ "your land" (and not mine or somebody's else)
land.akk.s.g. -

The possessive suffixes were in the singular
-ja "my"
-ki "thy",
-ka   "thy",fem.plk
-Su "his"
-Sa "her"
in the plural
-ni "our"
-kunu "your", mask. pl.
-kina "your",
-Sunu "their",
-Sina "their",

/-ja/ became d -i: (after consonants) or /-?a/ (after /-u:/ case endings) "my"

To give a framework, Akkadian was a SOV-language (though it could be
OSV to focus on the object) and used pronominal suffixes on verbs for
direct or indirect object. I did no research on other  languages with
possessive suffixes, but I think that a language with possessive
suffixes will also tend to use other pronominal suffixes.

All examples are from Kaspar K. Riemenschneiders "Lehrbuch des Akkadischen".