This week the LA River is once again open to kayak in. There's nothing quite like going through a major metropolitan waterway. You have now until Labor Day to go down the river, then it gets shut down again until next summer. You can enter two areas of the river the Sepulveda Basin (San Fernando Valley) and the Glendale Narrows (Elysian Valley). That's all you have access to, don't just plop in wherever you see the LA River, that'd be illegal.
The city has a terrible web-site for info about the rivers being open. What most people do is use a a service that provides kayaks to go down the river.