There are no gender-neutral pronouns that can be applied to a person. "Оно" has pretty much the same connotations as "it" in English. You can probably make something up, if you're familiar with how verbs and adjectives are formed in Russian, but I doubt it will be easy to read for a native speaker.
There are, however, ways to conceal one's gender in speech. That includes using passive verbs instead of active ones when it comes to past tense, but this is not always feasible (it can be done to a phrase like "я оплатила счёт", turning it into "счёт был оплачен мною", but good luck doing the same for stuff like "я вышел на улицу"). If the bulk of a text is written in past tense, you may try using present tense instead ("я вышел на улицу" becomes "я выхожу на улицу").
You can also refer to a person by name instead of using a third-person pronoun, but it's generally recommended that you avoid using tautology.
Some people use the so-called "gender gaps". The practice is controversial to say the least, as many find them confusing and hard to read. Basically you just put an underline after the gender-neutral part of the word is over, and then write a gender-specific ending, usually feminine, but usage of masculine is not unheard of. That way, the adjective "жёлтый/жёлтая" becomes "жёлт_ая" or "жёлт_ый". Thread with caution, however, as how acceptable its usage is largely depends on the audience it's meant for. If it's meant for the general population (and has nothing to do with feminism and so on), then I suggest using something else.
There's a different, but more accepted way of gender-neutral speech - putting the alternative ending in parentheses. "Грустный/грустная" can either be written as "грустный(ая)", "грустн(ый/ая)" (that's a bit weird though) or left as it is.
And some people just use masculine pronouns everywhere when they mean common gender. Some people do mind it, some don't.
Hope that helps.