Want to become more culturally aware, but would rather spend the $10 museum entrance fee on a movie?
Well, here's a list of San Francisco and East Bay museums and their free days. Now you don't have an excuse.

Are there museums with free or discounted days that I've missed? Let me know about it!!

African-American Museum and Library (Oakland)
Free admission
Tuesday - Saturday 12 - 5:30 p.m.

Asian Art Museum (San Francisco)
Free 1st Tuesday of the month.
Open Tuesday - Sunday, 10:00 a.m to 5:00 p.m. with extended evening
hours every Thursday until 9:00 p.m. (on Thursdays after 5 p.m.
admission is just $5 for all visitors except those under 12 and
members, who are always free.)

Badè Museum of Biblical Archaeology (Berkeley)
Admission is free
Hours: Tuesdays and Thursdays 10:00am - 3:00pm

Bay Area Discovery Museum (Sausalito)
The 2nd Saturday of each month is free for families 1 - 5 pm.
Regular Hours are
Tuesday - Friday, 9am - 4pm
Saturday & Sunday, 10am - 5pm
Closed Mondays, except when noted.

Berkeley Art Museum/Pacific Film Archive Theater (Berkeley)
Info for visiting can be found here

Cable Car Museum (San Francisco)
Free admission.
Hours: November-March, daily 10 AM to 5 PM; April-October daily 10 AM to 6 PM

California Academy of Sciences (San Francisco)
Free 1st Wednesday of the month
Hours of Operation: 10 am – 5 pm EVERY DAY OF THE YEAR except the Third Thursday of every month. On that day, the Academy is open until 9 pm. Admission on Third Thursday's is $5 for everyone.

California Indian Museum and Cultural Center (San Francisco)
"Recently, the California Indian Museum has moved its operations to
Santa Rosa, CA. The administration office is open Mon-Fri. from 9am
to 5pm. The Museum currently offers a free monthly lecture series, but
our exhibits are NOT open to the public until all renovations to our new
building are done."

Cartoon Art Museum (San Francisco)
The first Tuesday of every calendar month is "Pay What You Wish Day."
Open daily 11:00 - 5:00, Closed Mondays

Chabot Space & Science Center (Oakland)
Admission and hours are very complicated and require a slide rule to figure out. Check it out here

Chinese Historical Society of America Museum (San Francisco)
Free to the public on the first Thursday of every month
Hours: Tuesday through Friday 12:00pm-5:00pm
Saturday and Sunday 12:00pm-4:00pm

Contemporary Jewish Museum (San Francisco)
Free public tours on Sundays and Wednesday at 12:30pm.
Hours: Sunday through Thursday, 12pm-6pm.

De Young Museum (San Francisco)
First Tuesday of each month FREE
Hours: Tuesday through Sunday, 9:30 a.m. to 5:00 p.m.
Friday: Open until 8:45 p.m.
Muni riders with Fast Pass or transfer receive a $2 discount.

Ebony Museum of Art (Oakland)
Admission: Free (donations accepted)
Tue-Sat 11 a.m.-6 p.m. Closed Sun-Mon.

Exploratorium (San Francisco)
Free 1st Wednesday of the month.
Open Tuesday through Sunday.
10 A.M. to 5 P.M.

The Gold Rush Trail (San Francisco)
This appears to be absolutely free, since it's map of the historical districts of
San Franciso that you can walk at will. "The Gold Rush Trail runs from "South
of Market" (bottom of map) to the Embarcadero and back through downtown,
winding up in the historic Chinatown and North Beach neighborhoods: 49 sites
and key points of interest in six historic districts of San Francisco.

Habitot (Berkeley)
Free admission days are listed here
Other admission details listed here

Lawrence Hall of Science (Berkeley)
No free days. Check out hours and admission here

Legion of Honor (San Francisco)
Free every Tuesday.
Open Tues.–Sun. 9:30 a.m. to 5 p.m.
Muni riders with Fast Pass or transfer receive a $2 discount.

The Magnes Museum (Berkeley)
Does not appear to have any free days, but it looks interesting anyway, and
admission seems pretty cheap.
Sunday-Wednesday 11am-4pm
Thursday 11am-8pm

The Mexican Museum is still being built, but I'm sure it'll be
awesome when it's done.

Mills College Art Museum (Oakland)
Free admission
Hours: Tues 11:00 am–4:00 pm
Wed 11:00 am–7:30 pm
Thurs–Sat 11:00 am–4:00 pm
Sun 12:00 pm–4:00 pm

And the Musee Mechanique is always free, but the machines are coin-

Museum of Children's Art Oakland (Oakland)
Free admission
tuesday - friday 10am to 5pm
saturday - sunday noon to 5pm

Museum of Craft and Folk Art (San Francisco)
Enjoy free admission to the Museum on the first Tuesday of each month.
Hours: Tue – Fri: 11 am – 6 pm
Sat & Sun: 11 am – 5 pm

Museum of The African Diaspora (San Francisco)
No free days. Check out hours and admission here

Museum of the City of San Francisco (San Francisco)
This would appear to be limited to virtual exhibitions, which means admission
is the price of your internet connection. Still, looks cool and worth the time spent
UPDATE: turns out it does have an existence in real life:
Museum of the City of San Francisco
Free admission
Hours: Wedneday - Sunday, 10 AM to 6 PM

National Maritime Museum San Francisco (San Francisco)
Free admission to museum building
Hours: 10:00 am - 5:00 pm daily

Oakland Museum of CA (Oakland)
Free Second Sunday of each month
Hours: Wednesday through Saturday, 10 a.m. to 5 p.m
Sunday, noon to 5 p.m.
Closed Mondays and Tuesdays
First Friday of each month open until 9 p.m.

Oakland Western Aerospace Museum (Oakland)
Wednesday through Sunday 10 AM to 4 PM. (no free days, it would seem)

Pacific Heritage Museum (San Francisco)
Free admisssion
Hours: Tuesday to Saturday, 10:00 a.m. to 4:00 p.m.

Pardee Home Museum (Oakland)
No free days here, but they do have a variety of interesting tours available

Phoebe Hearst Museum of Anthropology (Berkeley)
Free to everyone on Thursdays!
Hours: 10:00 a.m.- 4:30 p.m. Wednesday - Saturday
12 p.m. - 4:00 p.m. on Sundays.

Randall Museum (San Francisco)
Free admission
Tuesday - Saturday, 10am to 5pm

Ripley's Believe it or Not Museum (San Francisco)
No free days
Hours: Sunday-Thursday 10 a.m. until 10 p.m.
Friday and Saturday 10 a.m. until midnight

San Francisco Airport Museums (San Francisco)
Free every day, you just have to get yourself to the airport. And now that BART will
be going there direct, it shouldn't be all that difficult.

San Francisco Fire Museum (San Francisco)
Free every day
Open Thursday through Sunday 1:00 p.m. to 4:00 p.m.

SFMOMA (San Francisco)
Free 1st Tuesday of the month.
Summer hours (Memorial Day through Labor Day):
Open every day (EXCEPT Wednesdays) 10 a.m.– 5:45 p.m. (On Thursday nights
6 - 9 p.m., admission is half price.)
Open every day (EXCEPT Wednesdays) 11 a.m.–5:45 p.m. (On Thursday nights
after 6 - 9 p.m., admission is half price.)

Strybing Arboretum (San Francisco)
This appears to be free, with free guided tours every day at 1:30 p.m.
The gardens are open daily, 365 days a year.
Weekdays, 8am to 4:30pm
Weekends & holidays, 10am to 5pm

Takara Sake USA Museum (Berkeley)
Free admission
Open daily from 12 pm to 6 pm

USS Hornet Museum (Alameda)
Admission info was a little confusing, check it out here

USS Potomac Oakland (Oakland)
No free rides here, but check site for interesting tour info

And there's Zeum, which doesn't appear to have free days, but it
seems pretty cheap, is geared towards kids, and sounds awfully cool.
School Year Hours: Open Wednesday–Sunday, 11 am–5 pm
Summer Hours: Open Tuesday–Sunday, 11 am–5 pm

Other resources for this sort of information

Sally's Place - SF Museums