If I'm not mistaken, the determiner DIŠ (which is literally just the sign for "one", a single cuneiform wedge) can sometimes be found also with female names. The double determiner DIŠ.MUNUS is also sometimes attested* for females, further suggesting that DIŠ was not always regarded as strongly male-specific.
Also, personal names were frequently written without any determiner at all,† especially in prose. For example, the usual Akkadian letter introduction formula‡ often doesn't feature any determiners at all. The use of determiners for personal names was more common in contexts such as witness lists for contracts, where it was important to unambiguously identify and separate consecutive names.
That said, determinatives or not, it's quite hard to say or write anything gender-neutral in Akkadian; like most Semitic languages, it has a strong binary gender distinction built deep into its grammar, with all nouns being either masculine or feminine, adjectives and numerals reflecting the gender of the noun they modify, most verb forms reflecting the gender of the subject and most pronouns — including both 2nd and 3rd person personal pronouns as well as demonstratives — being gendered.
In comparison, both Sumerian and Hittite have fairly gender-neutral grammars, with no grammatical masculine / feminine distinction or gendered pronouns. They do both have two "genders" or classes of nouns, which are often described as human / non-human, animate / inanimate or common / neuter. The first class (human / animate / common) includes all people, gods, etc., while the second (non-human / inanimate / neuter) tends to contain things like inanimate objects, mass nouns, abstract concepts, animals and groups of people.
(The Sumerian and Hittite noun class systems do differ in many details, and there's no reason that I know of to assume that their superficial similarity is anything but a curious coincidence. They do share some distinctive features, though, including the fact that in both systems only nouns of the first class can normally serve as actors, i.e. as the subject of a transitive verb, without employing some kind of a grammatical trick or circumlocution.)
*) This tablet (ARM 10,8; CDLI P349972) that I found with a quick CDLI search provides a nice illustrative example. Apparently a letter from queen Šibtu of Mari to her husband, king Zimri-Lim, it features the queen's own name written with the determinative MUNUS in the introduction, another female name Aḫātum written with DIŠ.MUNUS, the male names Dagan-Mālik and Zimri-Lim written with no determinative at all,† the feminine divine name Bēlet-ekallim ("Mistress of the Palace") written with the usual divine determinative DINGIR, and finally the male name Aḫum written with DIŠ. One wonders if perhaps the determinatives were added specifically to the names Aḫātum and Aḫum because they happen to also be common Akkadian nouns (aḫātum = sister, aḫum = brother) that could occur in a similar context, whereas there's no risk of misreading the names Zimri-Lim or Dagan-Mālik as anything but personal names where they occur. Although in this very similar letter (ARM 10,7; CDLI P349971), also from queen Šibtu to Zimri-Lim and also concerning a prophetic vision, Zimri-Lim's name is written with DIŠ, so… who knows?
†) That is, other than the determinative DINGIR used for divine names, which often ends up in Akkadian personal names since many of them start or end with the name of a deity, as in Dagan-Mālik = DINGIRda-gan-ma-lik in the aforementioned letter.
‡) a-na PN1 qí-bí-ma um-ma PN2-ma = "say to PN1, thus [says] PN2", as also seen in the letter mentioned above.