As a native English speaker, when I first started learning languages with gender at school, namely Spanish and German, the whole idea of nouns having gender seemed quite bizarre and, at the time, pointless. As I've studied other languages, I've begun to realise that having some sort of noun class is quite common and it no longer bothers me; in fact I quite like it at times. What I do still wonder about is the connection between grammatical genders in a language and gender ideas in society.
What should first be pointed out is that there is no connection between the existence of gender in a language and gender equality in society. We can look for example at Persian, a language which almost never shows gender -there is no difference between 'he' and 'she' and to explicitly show gender one must say 'man' or 'woman'- or at English which only shows gender in the personal and possessive pronouns (he/she/it, his/her/its). Without ignoring gender problems in the Anglosphere, I think it's fair to say that women there have it a lot better of than in the Islamic Republic of Iran. Similarly, German and Russian, both of which have 3 grammatical genders (masculine, feminine and neuter) for all nouns, belong to societies with quite different attitudes towards gender equality. One may point out that Icelandic has one of the highest uses of grammatical gender with not just 3 genders in the singular, but also in the plural (German and Russian, for example, have only one plural - Romance languages such as Spanish and French also have plural genders, but only 2). However, Icelandic is a very traditional language and has barely changed from that which was spoken by the Vikings hundreds of years ago, hardly the epitome of gender equality.
Despite that, there are still problems when dealing with languages with grammatical genders. Firstly we have the problem of how to talk about groups of men and women in Romance languages. Using Spanish as the example, when talking about a group of woman you say 'todas' - all, using the feminine form, when talking about a group of men you say 'todos', using the masculine form. When you want to refer to group of say two men and two women, you say 'todos', again using the masculine form, but it becomes worse because a group of three women and one man or even a hundred women and one man would be referred to as 'todos'. This is even the case when there are a couple of women and a male dog! Obviously you could say that the masculine 'todos' is just the gender neutral form, but there is no denying that, if this is the case, it derives from the simple masculine form. One attempt at overcoming this obvious bias is to use the '@' sign, which quite cleverly includes both the masculine 'o' and the feminine 'a', however, 'tod@s' would never make its way into official language because it looks too internet-y. Another system is to use 'x' as in 'todxs', but this has the added problem of not reflecting the pronunciation at all. Furthermore, these systems are sometimes also used in Catalan, but have less effect since Catalan often has no ending for the masculine plural, so all becomes 'tots/totes' and seeing 'tot@s/totxs' looks like a direct borrowing from Spanish (which is a whole other issue).
I personally don't see this changing any time soon since any forced language change would be aggressively rejected considering how much it would have to change the grammatical structure (imagine being forced to say 'hes' and 'shes' instead of 'they'). That said, I wouldn't mind seeing something like 'todes' or 'todis' coming into use, although this would, in all likelihood, seem ridiculous to a native speaker.
A second problem is how to deal with using masculine nouns to refer to professions or groups which can be either gender. This issue doesn't arise much in English since the word 'doctor', for example, has no obvious gender, however, in gendered languages the base form is often masculine -not surprising considering the relatively recent arrival of lots of women to the profession- and the female form is derived from it: e.g. German - der Arzt (m.) / die Arztin (f.), Catalan - el metge (m.) / la metgessa (f.) and even though in Spanish the final vowel simply changes (médico(m.)/médica(f.)) meaning that the feminine form isn't necessarily a derivation, the masculine form is still the standard and the female form is only used if you know that the doctor is actually a women (you would say 'voy al médico' - I'm going to the doctor, even if it turns out that the doctor is a woman).
In German there is a small trend where people might use the feminine form, especially in the plural, as the gender-neutral form. Thus, 'AktivistInnen' (with capital 'I') means activists of both genders. Although there is still the problem that the feminine form is only a derivation of the masculine form, as in you have to add or change the masculine form to get the feminine form. This can also be seen in English with words like 'actor' and 'actress', although it's not as common.
A final point on English: when referring to a person, but not wanting to specify the gender, you have a few possibilities. One is to use 'they' which has the meaning of 'that person' when used in singular, however, this has gained a colloquial connotation despite being a very old usage of the word. You can sometimes use 'one', but this is often too formal. It is therefore quite common to see someone using 'he' to refer to anyone when the gender is unimportant, especially in writing. That said, it is nice and refreshing to see a writer use 'she' instead - especially when the topic has nothing to do with gender studies. Now before the anti-feminists whip themselves into a rage, this doesn't equate to misandry since this is still extremely rare.
Finally, it is worth noting that all of the attempts to mitigate gender bias in language which I have discussed above have only a limited usage and represent the more open-minded parts of society. Thus I don't think it is worthwhile trying to change language usage in order to affect attitudes, but rather try to educate people and then let the language change organically as people decide to speak in a more fair and equal manner.