The Art Institute is free every Thursday right now between 5:00pm and 8:00pm. Make sure you check back on their site or call them;I am pretty sure it will still be free in March on Thursdays during this time frame.
The Lincoln Park Zoo is free year around.
The Adler Planetarium is free all day on these days in March : 3, 10 & 17. (Additional fees for shows, tours and the Atwood Sphere Experience apply.)
The Natural History Museum is free on March 9th.
The Shedd Aquarium has not announced its discount days for 2009 yet:
"On our Community Discount Days, general admission is free, and packages including Wild Reef are available at a discounted rate. Shedd's 52 discount days a year take advantage of civic and school holidays to make family visits more convenient.
There are no remaining discount days in 2008. Community Discount Days resume in 2009 the week of Jan. 10 through 16, and continue on Mondays and Tuesdays in January beginning on Jan. 19. Watch for our discount schedule for all of 2009."
http://www.sheddaquarium.org/discounts.html
The Museum of Contemporary Art is free every Tuesday.
The Museum of Science and Industry has no free days in March.