Being a tourist, can I film police in the US?

There is no law against filming police in action, anywhere in the USA that I am aware of. But to be honest, sticking your cellphone in the officers face to record what he says will likely not earn you any leniency. So don't be surprised if you get a ticket and not a warning.

Police as a whole in the USA are fine but stern. Officers that act like what you see on YuckTube are a minority and you would be hard pressed to find a police force anywhere in the world without a few bad apples.

One drawback to social media is that incidents get blown up to blanket accusations against all, when they should be viewed as individual acts. Story telling in days of old had similar results, as a story was told by subsequent people it became more fabulous or more dastardly. Back then it took years for stories to become legends, now with social media it takes hours. But the end result is the same, the legend doesn't really reflect the act.

You have the right to film police, although police do not always agree. Relevant article. Note that most Constitutional rights apply to everyone in the country, not just citizens. That is, you can't be tried twice for the same offense just because you are a tourist. (Disclaimer 1: You can be tried for essentially the same offense in one State court and Federal court; Disclaimer 2: IANAL)

Legally, yes, as others have explained.

Practically, it's not a good idea. US citizens are routinely arrested for filming police actions (even though most are never convicted or even indicted) and their videos confiscated. Since you're a non-citizen, having an arrest record in the US will likely make you ineligible to enter the US again.

Whether that's worth the risk is something only you can decide.

Taking silent film or still photos:
As far as I know, there are no laws prohibiting you taking still photos or silent films of police officers in public lands or roads. In private places (including museums, libraries or schools) the owner of the property or related authorities may have their own rules, for instance, stores prohibiting taking photos of their employees, or an airport prohibiting photo taking at security checkpoint.

However, a police officer in public places has the right to order you (citizen or not) to stop taking photos or films when he/she believes that you are interfering their job. Even if you believe it's not the case, you should never resist his/her order or try to run away from the scene.

Recording audio or films containing audio:
In some states, recording audio (or a film with audio) of police officers may not be legal. Also keep in mind that photos/films legally taken may not be legally published online.