Make your antecedents plural.
Refer to people, not a person. Refer to readers, not a reader. Talk about neighbors, not a neighbor. Then you can use they, their-theirs, them, and themselves.
Some texts now use this approach. When generalizing, they make antecedents plural. They can then use third-person plural pronouns, which do not reveal gender:
Veterinarians are encouraged to modify and adapt the sample forms included here to fit their needs.
James F. Wilson, Law and Ethics of the Veterinary Profession, p. 248 (1990).
That’s the solution I use. Whenever I’m writing about a generic type of person, I refer instead to people. Then I can use they, their-theirs, them, and themselves.
Read the next topic for more solutions to sexist writing.