If you believe in FDIC coverage and you're under the limits, you probably don't need to do anything. In fact, you might even speed or cause the bank's demise if you pull out in fear of a potential bank failure (by participating in a bank run). On the other hand, maybe it's less hassle to go through the account opening process at a new bank than it would be to deal with an FDIC takeover - who knows?
The Consumerist today made a plea for customers to keep their cool in spite of all the hype. The resulting debates in the comments illustrate the pros and cons of keeping your WaMu accounts open - even if they might fail. See Don't Start Yanking Your WaMu Accounts for more information.
Further reading:


I understand that my money is FDIC insured, I can still write checks as before and that JP Morgan has bought “the deposits” from Washington Mutual. In general things will continue much the same.
However, towards the end of WaMu life they were offering great CD rates at 5% APY. When will we know if JP Morgan will agree to honor those rates? The FDIC Q&A leaves that unanswered. http://www.fdic.gov/bank/individual/failed/wamu_q_and_a.html#interest
I was under the impression that if you lock it in at a 5.00% APY for the year, the rate is the same for the entire term (unless it was sold as a variable-rate CD). However, the rates of WaMu’s online savings account could be subject to change at any time (currently 4.00% APY).
Although it’s probably way too early to tell, it certainly wouldn’t surprise me if Chase decided to offer lower rates for future CD or savings accounts. I imagine that if Chase played with the interest rates too much, they may risk losing a significant number of deposits if a large percentage of WaMu’s $188 billion in deposits were high-yield savings & CD accounts.
I recently did another post with my understanding of the answer: Will WaMu Rates Stand After Failure?