The restrictiveness of your sprites are reminiscent of some RPGs on the NES.
What are you making these sprites for? It's not fair that you just post the sprites and demand quick c+c!
As of now, your sprites look kind of monochromatic. That can be a good thing, but with what you're trying to do, a little color enhancement can mean a whole lot.
The skin tone you have chosen is very subtle and washed out next to white. The black in your palette is too strong against these colors and drowns all the detail. If you're going to stick to three colors, you will need to keep a dark color. I would experiment a little with heavily saturated dark colors. In any case, your second color should be somewhere in between that and your lightest color. Since your palette is so small, try to make a "ramp" between your darkest and lightest colors with the one in the middle.
Many sprite palettes on the NES do not use the peach skin tone that you are using here, just FYI. If you're going to stick with it, instead of using white, you may want to use a very washed out sky blue, which will stand out against the orange-y skin tone (basic color theory; complementary colors).
For the most part, your sprite animation is OK. If I saw it in a NES game, it would probably fit right in. Not much you can do with a limited frame rate and the size limitation, but you might consider animating his hair a little bit for a final touch.