What's the trick to earning enough points in Mayhem activities?

In the Mayhem levels, the trick is to get a high multiplier and keep it there. You get credit for destroying almost EVERYTHING, so driving/running down the street and firing a RPG or the tank cannon repeatedly into the sidewalk is a good way to go. If you can find a roadside cafe, a street market, or a line of fences, you're golden. Alleyways are positively littered with things that explode for cash.

It's relatively easy to hit 30-50+ multiplier in non-Tank Mayhem, which means you'll be racking up the score in no time. If your multiplier is super high, target expensive cars to get a huge points bonus.

The most important thing though, is just keep shooting. You should be firing off rounds as fast as you possibly can, and blowing absolutely everything in sight to kingdom come.

It helps to move into a target-rich area, like a suburban street where fences and small items litter the scenery. You're almost better running (or stealing a car) and driving to a better part of town than staying where there's not much to shoot at. The time you lose is far outweighed by the extra multiplier you will gain. I haven't extensively scouted the map to find the best place yet, but you might want to take a minute to drive around and look at object placement before you start a level, so you can plan your route better.

Don't bother with guns in this mode - if you have a RPG, great, but watch for splash damage that can knock you down and end your streak. Alternatively, use thrown weapons (like grenades, for example) which can take a whole area out. You can generally throw these super fast, and your throwing distance is generally far enough that you won't be in the blast radius when they go off. Run down the street and just huck them in random directions towards sidewalks, cars, etc. If you're having trouble destroying things with grenades, don't forget that you can upgrade them at the weapon store to make them cause damage in a greater range, and allow you to throw them further.

Mayhem is my second favorite activity, after Insurance Fraud. Enjoy it! :)

Drive your tank to the Mayhem level, get out and start the level. Your tank should still be there, so get back in and let loose.

Agent86 touched on it, but it deserves highlighting: GRENADES.

In the Mayhem levels that don't give you an explosive weapon, you'll still have your grenades. Start throwing them at anything and everything, and you'll eventually annoy the gangs and/or police. Build up your wanted level enough, and you'll complete the Mayhem mission via destroyed cop cars and gang cars.