While you won't lose something essential to the chassis or suspension just driving down the road, suspicious noises should be checked out as soon as they are heard. Plus, who wants to drive an annoying clanking vehicle?
Unfortunately, finding the cause of a vehicle's quirky noise isn't so easy. The complex nature of modern suspensions and systems may distort the sound as it travels through the body, making it hard to zero in on.
Even though a sound comes from under a car it may have nothing to do with the suspension or steering systems. It takes a skilled ear to know the difference. Take the exhaust system, for example. If everything—muffler, exhaust pipes, tailpipe, and the catalytic converter—isn't positioned just right, or the hangers are loose or broken, the broken contact will result in a clunking between these components and the chassis or drive shaft.
A broken motor mount can also cause a solid thump. A common cause of this is oil that is left soaking, causing it to de-laminate, or perhaps the bolts just came loose. In this case, it will be noticeable when getting on or off the throttle but won't occur when driving over bumps.
Typically, the only real fix is replacement, and may be more intensive and expensive than you might expect. For instance, a rusted out suspension makes a terrible noise and needs welding or complete replacement; that's a problem that can't be ignored for a second.
On the other hand, if you do your detective work correctly, you may save a lot of money by finding an easily fix culprit, like a strut gland nut that needs tightening. A loose upper strut cartridge nut will cause an especially hard-to-find rattle but, once diagnosed, is easy as pie to fix.