I now change the battery before it dies. Reports are that  use of older
batteries may shorten the alternator's life. See below for citations on this.

My 91 Civic, 158k miles, is on its fourth battery and second alternator. The
first two batteries each died after 4.5 years. The car was driven in mostly
Northern climates. Each had a few jumps performed on them, which I understand
tends to shorten battery life. The alternator died after eight years.

Businesses like Sears, dealerships, and independent shops offer battery checks
for a small fee. Autozone will do a battery check for free but I'm not sure I'd
rely on what Autozone's "experts" find. Still, it's no charge. Google on the net
for details on what these checks include.

I keep an eye on how the car is cranking at cold startup. With experience, one
can often tell when the the battery is near-death. I also have benchmark
readings of the battery voltage with the car running, with headlights on and not
running, everything off.

Three years seems a little soon. Do you have any symptoms of the battery going
bad? Had many jump starts on it? IIRC, some have reported getting as much as
seven years ou of their Honda battery.

Re dying batteries damaging the alternator:

"A weak battery that is not storing enough power will cause the alternator to
work harder and possibly cause premature alternator failure. Dirty or corroded
battery terminals can severely reduce the lifespan of the battery and
alternator."  http://www.trustmymechanic.com/35.html

" ...depleted batteries make recharging much more difficult, shortening
alternator life." http://www.intra-tech.com/news/homeapriltwentnine.html

"By maintaining batteries in peak condition alternators do not work as hard,
batteries accept charge more readily so alternator life is extended. Healthy
batteries start engines easier, so your starter should also last longer."
http://www.megapulse.net/faq.htm

Do what you think is prudent for your situation.  My 99 Civic's battery was
showing signs of going bad 2 months ago....slower slightly labored cranking
than normal for startup...so I went to Costco and replaced it for $36.  The
original battery lasted just over 5 yrs, the longest lasting battery I've
ever had.

One of the problems when changing batteries is losing all the settings on
your radio.  I get around that problem by connecting a 9V battery across the
car battery cables until I get the new battery in place and the cables
reconnected.  Works beautifully.

Bearman

Should be too early.  My 98 Civic could be getting close, but less
than 4 years is too early.