You can. If you set keys directly on maya objects over time , it'll animate.
The chance that you'll be effective working like that, however, is very, very low. The number of keys you need to set to describe a even a very simple motion in nothing but direct translate or rotate keys is much, much larger - at least one and maybe two orders of magnitude larger.
The whole job of the rig is to tell the computer how to get from pose to pose so you can concentrate on the look of the poses and the timings. Rigs are the tech that do what you're looking for. You may need to shop around for a better, more user-friendly rig (maybe people can chime in with free examples or youtubes that show good rigs) . But working without rigs isn't a good idea -- there's a reason that nobody who animates for a living will do that voluntarily!