A friend tells me that there is a cleaning solution available which
stands overnight in sink drains to clear away grease build-up.

I've settle on a non-chemical fix:
I dismantle the drains, pull out the dead-rat looking mass of hair and
soap scum and then put the drains back together.
It takes about a half hour and I know that the drain is perfectly
clear until next time.
No chemicals and no cost other than my time.

We're on a private septic system, so we try not to pour chemicals down our
drains.

Like you, we simply pull the trap assemblies apart a couple times a year
and clean them out. The vast majority of the hair and gunk is within the
first foot or so of the drains, so this gets us free flowing pipes again.

My wife actually takes ours apart and cleans them out. She thinks it's
"fun". Saves me the messy job.

Anthony