As a way of getting gendered pronouns out of the language I think pushing the boundaries of singular they is a good approach. But the final step of saying "Pat doesn't know their way home" is still too strange for many people [1]. One way to avoid this is to substitute articles for pronouns:

1.a	Pat doesn't know his way home.
1.b	Pat doesn't know the way home.
2.a	Sam is careless. Keep that in mind when evaluating her paper.
2.a	Sam is careless. Keep that in mind when evaluating the paper.

In many cases the pronoun isn't indicating much that people won't be able to get from context, or the additional information isn't very important. Sometimes whether it's important depends on context:

3.a	Bill passed his laptop to Mike.
3.b	Bill passed the laptop to Mike.

Here we lose that the laptop belongs to Bill. If this is significant in the context then it's probably not worth avoiding the pronoun.

You do need to be careful; in many cases this can make you sound like English is your second language:

4.a	Alex is wearing his jacket inside out.
4.b	Alex is wearing the jacket inside out.
5.a	Leslie is brushing her teeth.
5.b	Leslie is brushing the teeth.

This may not be a problem, but is something to be aware of.

This is something I can only use occasionally, but I do like having another way to avoid bringing in gender when it's completely irrelevant.

Update 2013-11-27: I noticed LinkedIn using this today, with emails titled "Congratulate [name] on the new job".

[1] Including me. But here's an example I ran across this morning:

Janani is a South Asian electron spinning around the Bay Area making art and scholarship. They're newly the Assistant Editor of Black Girl Dangerous! You can read more of their work at queerdarkenergy.sqsp.com, and catch their poetry tour through New England at bit.ly/queerdarkmatter.   --  source